<PAGE>   1
 
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                       SECURITIES AND EXCHANGE COMMISSION
                            WASHINGTON, D. C. 20549
                                   FORM 10-Q
                   QUARTERLY REPORT UNDER SECTION 13 OR 15(D)
                     OF THE SECURITIES EXCHANGE ACT OF 1934
 
(Mark One)
 
[X] QUARTERLY REPORT PURSUANT TO SECTION 13 or 15(d) OF THE SECURITIES EXCHANGE
    ACT OF 1934
 
FOR THE QUARTERLY PERIOD ENDED JUNE 30, 1998
 
                                       OR
 
[ ] TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE
    ACT OF 1934
 
For the transition period from ____________ to ____________
 
Commission file number 1-10319
 
                              RMI TITANIUM COMPANY
             (Exact name of registrant as specified in its charter)
 
<TABLE>
<S>                                            <C>
             OHIO                                  31-0875005
(State or other jurisdiction of                 (I.R.S. Employer
 incorporation or organization)                Identification No.)
</TABLE>
 
                     1000 WARREN AVENUE, NILES, OHIO 44446
                    (Address of principal executive offices)
 
                                 (330) 544-7700
              (Registrant's telephone number, including area code)
 
     Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15 (d) of the Securities Exchange Act of
1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days.
 
                                  YES X NO  __
 
     At August 1, 1998, 20,521,302 shares of common stock of the registrant were
outstanding.
 
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<PAGE>   2
 
                              RMI TITANIUM COMPANY
                                   FORM 10-Q
                          QUARTER ENDED JUNE 30, 1998
 
                                     INDEX
 
<TABLE>
<CAPTION>
                                                              PAGE
                                                              ----
<S>                                                           <C>
PART I--FINANCIAL INFORMATION
Item 1. Consolidated Financial Statements:
     Introduction to Consolidated Financial Statements......    2
     Consolidated Statement of Income.......................    3
     Consolidated Balance Sheet.............................    4
     Consolidated Statement of Cash Flows...................    5
     Consolidated Statement of Changes in Shareholders'
      Equity................................................    6
     Selected Notes to Consolidated Financial Statements....    7
Item 2. Management's Discussion and Analysis of Results of
  Operations and Financial Condition........................    9
PART II--OTHER INFORMATION
Item 6. Exhibits and Reports on Form 8-K....................   16
Signatures..................................................   17
</TABLE>
<PAGE>   3
 
                         PART I--FINANCIAL INFORMATION
 
ITEM 1. CONSOLIDATED FINANCIAL STATEMENTS
 
               INTRODUCTION TO CONSOLIDATED FINANCIAL STATEMENTS
 
     The consolidated financial statements included herein have been prepared by
RMI Titanium Company (the "Company"), without audit, pursuant to the rules and
regulations of the Securities and Exchange Commission. The financial information
presented reflects all adjustments, consisting only of normal recurring
adjustments, which are, in the opinion of management, necessary for a fair
statement of the results for the interim periods presented. The results for the
interim periods are not necessarily indicative of the results to be expected for
the year.
 
                                        2
<PAGE>   4
 
                              RMI TITANIUM COMPANY
 
                        CONSOLIDATED STATEMENT OF INCOME
                                  (UNAUDITED)
 
                             (DOLLARS IN THOUSANDS)
 
<TABLE>
<CAPTION>
                                                QUARTER ENDED                 SIX MONTHS ENDED
                                                   JUNE 30                         JUNE 30
                                          -------------------------       -------------------------
                                             1998          1997              1998          1997
                                             ----          ----              ----          ----
<S>                                       <C>           <C>               <C>           <C>
Sales...................................  $    96,542   $    75,034       $   185,581   $   148,742
Operating costs:
Cost of sales...........................       72,364        58,313           135,885       116,245
Selling, general and administrative
  expenses..............................        3,810         2,805             7,816         5,531
Research, technical and product
  development expenses..................          974           698             1,821         1,402
                                          -----------   -----------       -----------   -----------
     Total operating costs..............       77,148        61,816           145,522       123,178
Operating income........................       19,394        13,218            40,059        25,564
Other income-net........................          881           261             1,423           426
Interest expense........................           (3)          (26)              (57)          (59)
                                          -----------   -----------       -----------   -----------
Income before income taxes..............       20,272        13,453            41,425        25,931
Provision for income taxes (Note 3).....      (10,138)        1,363            (4,038)        2,923
                                          -----------   -----------       -----------   -----------
Net income..............................  $    30,410   $    12,090       $    45,463   $    23,008
                                          ===========   ===========       ===========   ===========
Net income per common share:
  Basic.................................  $      1.48   $      0.59       $      2.22   $      1.13
                                          ===========   ===========       ===========   ===========
  Diluted...............................  $      1.47   $      0.57       $      2.20   $      1.12
                                          ===========   ===========       ===========   ===========
Weighted average shares outstanding:
  Basic.................................   20,512,420    20,393,337        20,497,628    20,336,777
                                          ===========   ===========       ===========   ===========
  Diluted...............................   20,645,859    20,847,801        20,645,859    20,560,296
                                          ===========   ===========       ===========   ===========
</TABLE>
 
  The accompanying notes are an integral part of these Consolidated Financial
                                  Statements.
 
                                        3
<PAGE>   5
 
                              RMI TITANIUM COMPANY
 
                           CONSOLIDATED BALANCE SHEET
                             (DOLLARS IN THOUSANDS)
 
<TABLE>
<CAPTION>
                                                                JUNE 30     DECEMBER 31
                                                                 1998          1997
                                                              -----------   -----------
                                                              (UNAUDITED)
<S>                                                           <C>           <C>
ASSETS
Current assets:
Cash and cash equivalents...................................   $ 40,705      $ 30,211
  Receivables--less allowance for doubtful accounts of
     $1,327 and $1,064......................................     66,688        70,898
  Inventories (Note 5)......................................    138,989       120,732
  Deferred tax asset........................................      1,211         4,811
  Other current assets......................................      5,675         5,903
                                                               --------      --------
     Total current assets...................................    253,268       232,555
  Property, plant and equipment, net of accumulated
     depreciation...........................................     51,589        43,034
  Noncurrent deferred tax asset.............................     21,209         5,157
  Other noncurrent assets...................................     11,467        10,563
                                                               --------      --------
     Total assets...........................................   $337,533      $291,309
                                                               ========      ========
LIABILITIES AND SHAREHOLDERS' EQUITY
Current liabilities:
  Accounts payable..........................................   $ 21,495      $ 25,346
  Accrued wages and other employee costs....................      9,807         8,024
  Other accrued liabilities.................................     16,746        14,361
                                                               --------      --------
     Total current liabilities..............................     48,048        47,731
Long-term debt (Note 6).....................................         --            --
Accrued postretirement benefit cost.........................     19,376        19,376
Other noncurrent liabilities................................      3,202         3,029
                                                               --------      --------
     Total liabilities......................................     70,626        70,136
                                                               --------      --------
Contingencies (Note 4)
Shareholders' equity:
  Preferred Stock, no par value; 5,000,000 shares
     authorized; no shares outstanding......................         --            --
  Common Stock, $0.01 par value, 30,000,000 shares
     authorized; 21,093,603 and 21,022,253 shares issued....        211           210
  Additional paid-in capital................................    237,996       236,970
  Accumulated earnings (deficit)............................     33,781       (11,682)
  Deferred compensation.....................................     (1,819)       (1,100)
  Treasury Common Stock at cost 577,301 and 575,485
     shares.................................................     (3,262)       (3,225)
                                                               --------      --------
Total shareholders' equity..................................    266,907       221,173
                                                               --------      --------
     Total liabilities and shareholders' equity.............   $337,533      $291,309
                                                               ========      ========
</TABLE>
 
  The accompanying notes are an integral part of these Consolidated Financial
                                  Statements.
 
                                        4
<PAGE>   6
 
                              RMI TITANIUM COMPANY
 
                      CONSOLIDATED STATEMENT OF CASH FLOWS
                                  (UNAUDITED)
 
                             (DOLLARS IN THOUSANDS)
 
<TABLE>
<CAPTION>
                                                              SIX MONTHS ENDED
                                                                   JUNE 30
                                                              -----------------
                                                               1998      1997
                                                               ----      ----
<S>                                                           <C>       <C>
CASH FLOWS FROM OPERATING ACTIVITIES:
Net income..................................................  $45,463   $23,008
Adjustment for items not affecting funds from operations:
  Depreciation..............................................    2,724     2,520
  Deferred taxes............................................  (12,452)    2,923
  Other-net.................................................      200       256
                                                              -------   -------
                                                               35,935    28,707
                                                              -------   -------
CHANGES IN ASSETS AND LIABILITIES (EXCLUDING CASH):
Receivables.................................................    4,210    (5,665)
Inventories.................................................  (18,257)   (8,770)
Accounts payable............................................   (3,851)    1,012
Other liabilities...........................................    4,341       196
Other assets................................................     (676)      394
                                                              -------   -------
                                                              (14,233)  (12,833)
                                                              -------   -------
     Cash provided by operating activities..................   21,702    15,874
                                                              -------   -------
CASH FLOWS FROM INVESTING ACTIVITIES:
  Capital expenditures......................................  (11,279)   (1,854)
                                                              -------   -------
     Cash used in investing activities......................  (11,279)   (1,854)
                                                              -------   -------
CASH FLOWS FROM FINANCING ACTIVITIES:
  Proceeds from issuance of Common Stock....................      108       929
  Debt repayments...........................................       --    (2,960)
  Treasury Common Stock purchased...........................      (37)     (147)
                                                              -------   -------
  Cash provided by (used in) financing activities...........       71    (2,178)
                                                              -------   -------
INCREASE IN CASH AND CASH EQUIVALENTS.......................   10,494    11,842
CASH AND CASH EQUIVALENTS AT BEGINNING OF PERIOD............  $30,211   $ 5,944
                                                              -------   -------
CASH AND CASH EQUIVALENTS AT END OF PERIOD..................  $40,705   $17,786
                                                              =======   =======
SUPPLEMENTAL CASH FLOW INFORMATION:
  Cash paid for
     Interest (net of amounts capitalized)..................  $    20   $    28
     Income taxes...........................................  $ 5,000   $   633
Noncash financing activities
     Issuance of restricted stock...........................  $   919   $   832
</TABLE>
 
  The accompanying notes are an integral part of these Consolidated Financial
                                  Statements.
 
                                        5
<PAGE>   7
 
                              RMI TITANIUM COMPANY
 
           CONSOLIDATED STATEMENT OF CHANGES IN SHAREHOLDERS' EQUITY
                                  (UNAUDITED)
 
                             (DOLLARS IN THOUSANDS)
 
<TABLE>
<CAPTION>
                                                                            ADDT'L.    RETAINED    TREASURY
                                        SHARES      COMMON     DEFERRED     PAID-IN    EARNINGS     COMMON    COMPREHENSIVE
                                      OUTSTANDING   STOCK    COMPENSATION   CAPITAL    (DEFICIT)    STOCK        INCOME
                                      -----------   -----    ------------   -------    ---------    -----        ------
<S>                                   <C>           <C>      <C>            <C>        <C>         <C>        <C>
Balance at December 31, 1997........  20,446,768     $210      $(1,100)     $236,970   $(11,682)   $(3,225)      $    --
Shares issued for Directors'
  Compensation......................          --       --           --            --         --         --            --
Shares issued for Restricted Stock
  Award Plans.......................      45,125       --         (919)          919         --         --            --
Compensation expense recognized.....          --       --          200            --         --         --            --
Treasury Common Stock purchased at
  cost..............................      (1,816)      --           --            --         --        (37)           --
Shares issued from exercise of
  employee stock options............      26,225        1           --           107         --         --            --
Net income..........................          --       --           --            --     45,463         --        45,463
  Other comprehensive income........          --       --           --            --         --         --            --
                                                                                                                 -------
  Comprehensive income..............          --       --           --            --         --         --       $45,463
                                      ----------     ----      -------      --------   --------    -------       =======
Balance at June 30, 1998............  20,516,302     $211      $(1,819)     $237,996   $ 33,781    $(3,262)
                                      ==========     ====      =======      ========   ========    =======
</TABLE>
 
  The accompanying notes are an integral part of these Consolidated Financial
                                  Statements.
 
                                        6
<PAGE>   8
 
                              RMI TITANIUM COMPANY
 
              SELECTED NOTES TO CONSOLIDATED FINANCIAL STATEMENTS
                                  (UNAUDITED)
 
NOTE 1--GENERAL
 
     The consolidated financial statements include the accounts of RMI Titanium
Company and its majority owned subsidiaries. All significant intercompany
transactions are eliminated. The Company's operations are conducted in one
business segment, the production and marketing of titanium metal and related
products.
 
     On July 3, 1997, the Company acquired 90% of the common stock of Galt
Alloys, Inc., a manufacturer of ferrotitanium and a producer and worldwide
distributor of specialty alloys to ferrous and nonferrous customers. RMI's
investment in Galt amounts to an initial cash investment of $3 million and an
agreement to invest up to an additional $17 million to finance a major expansion
program at Galt, which is expected to be completed in mid 1999. Through June 30,
1998, approximately $9.1 million had been expended on the expansion. Results of
Galt's operations are consolidated with RMI as of the acquisition date.
 
NOTE 2--ORGANIZATION
 
     The Company is a successor to entities that have been operating in the
titanium industry since 1958. In 1990, USX Corporation ("USX") and Quantum
Chemical Corporation ("Quantum") transferred their entire ownership interest in
the Company's immediate predecessor, RMI Company, an Ohio general partnership,
to the Company in exchange for shares of the Company's Common Stock (the
"Reorganization"). Quantum then sold its shares to the public. USX retained
ownership of its shares. At June 30, 1998 approximately 27% of the outstanding
common stock was owned by USX.
 
     In November, 1996, USX Corporation completed a public offering of its notes
which are exchangeable in February, 2000, for 5,483,600 shares of RMI Common
Stock, owned by USX (or for an equivalent amount of cash at USX's option). Such
shares represent all of the RMI Common Stock owned by USX.
 
NOTE 3--INCOME TAXES
 
     In the three and six month periods ended June 30, 1998 the Company recorded
an income tax benefit of $10.1 million and $4.0 million, respectively. These
amounts are comprised of an income tax provision against pretax income for the
three month period of $5.9 million and the six month period of $12.0 million and
an income tax benefit of $16.1 million for both periods, resulting from an
adjustment to reduce the deferred tax asset valuation allowance due to changes
in the Company's expectations about the ultimate realization of its deferred tax
assets. Excluding the $16.1 million valuation allowance adjustment, the
effective tax rate for the three months and the six months ended June 30, 1998
was approximately 29.0 percent. Because of available net operating loss
carryforwards, the amount of current taxes expected to be paid in 1998 will be
well below the level which could be expected by applying the statutory federal
tax rate of 35% to pretax income.
 
     Statement of Financial Accounting Standards No. 109 (SFAS No. 109),
"Accounting for Income Taxes," requires a valuation allowance when it is "more
likely than not" that some portion or all of the deferred tax assets will not be
realized. It further states that "forming a conclusion that a valuation
allowance is not needed is difficult when there is negative evidence such as
cumulative losses in recent years." The ultimate realization of this deferred
income tax asset depends upon the Company's ability to generate sufficient
taxable income prior to the expiration of its loss carryforwards. The Company
has evaluated the available evidence supporting the realization of future
taxable income and, based upon that evaluation, believes it is more likely than
not that its deferred tax assets will be realized. Based on that evaluation, the
valuation allowance was reduced.
 
                                        7
<PAGE>   9
 
NOTE 4--CONTINGENCIES
 
     In the ordinary course of business, the Company is subject to environmental
laws and regulations concerning the production, handling, storage,
transportation, emission, and disposal of waste materials and is also subject to
other federal and state laws and regulations regarding health and safety
matters. These laws and regulations are constantly evolving, and it is not
currently possible to predict accurately the ultimate effect these laws and
regulations will have on the Company in the future.
 
     The Company is involved in investigative or cleanup projects under federal
or state environmental laws at a number of waste disposal sites, including the
Fields Brook Superfund Site. Given the status of the proceedings with respect to
these sites, ultimate investigative and remediation costs cannot presently be
accurately predicted, but could, in the aggregate, be material. Based on the
information available regarding the current ranges of estimated remediation
costs at currently active sites, and what the Company believes will be its
ultimate share of such costs, provisions for environmental-related costs have
been recorded. These provisions are in addition to amounts which have previously
been accrued for the Company's share of environmental study costs.
 
     With regard to the Fields Brook Superfund Site, the Company, together with
31 other companies, has been identified by the EPA as a potentially responsible
party ("PRP") under the Comprehensive Environmental Response, Compensation
Liability Act ("CERCLA") with respect to a superfund site defined as the Fields
Brook Watershed in Ashtabula, Ohio, which includes the Company's now closed
Ashtabula facilities. The EPA's 1986 estimate of the cost of remediation of the
Fields Brook operable sediment unit was $48 million. Recent studies, together
with improved remediation technology and redefined cleanup standards have
resulted in a more recent estimate of the remediation cost of approximately $25
million. The actual cost of remediation may vary from the estimate depending
upon any number of factors.
 
     The EPA, beginning in March 1989, ordered 22 of the PRPs to conduct a
design phase study for the sediment operable unit and a source control study.
These studies are nearly complete. The Company, working cooperatively with
fourteen other PRPs in accordance with two separate agreements, is complying
with the order. The Company has accrued and has been paying its portion of the
cost of such compliance. Actual cleanup will not commence prior to 1999. The
Company's share of the study costs has been established at 9.95%. In June, 1995,
the Company and twelve other PRPs entered into a Phase 2 (actual cleanup)
allocation agreement which assigns 9.44% of the cost to RMI. However, the actual
percentage may be more or less based on contributions from other parties which
are not currently participating in the Phase 2 allocation agreement.
 
     At June 30, 1998, the amount accrued for future environmental-related costs
was $3.5 million. Based on available information, RMI believes its share of
potential environmental-related costs, before expected contributions from third
parties, is in a range from $4.2 million to $7.0 million, in the aggregate. The
amount accrued is net of expected contributions from third parties (which does
not include any amounts from insurers) of approximately $2.1 million, which the
Company believes are probable. The Company has been receiving contributions from
such third parties for a number of years as partial reimbursement for costs
incurred by the Company. As these proceedings continue toward final resolution,
amounts in excess of those already provided may be necessary to discharge the
Company from its obligations for these projects.
 
     The ultimate resolution of these foregoing contingencies could,
individually or in the aggregate, be material to the consolidated financial
statements. However, management believes that the Company will remain a viable
and competitive enterprise even though it is possible that these matters could
be resolved unfavorably.
 
                                        8
<PAGE>   10
 
NOTE 5--INVENTORIES:
 
<TABLE>
<CAPTION>
                                                       (DOLLARS IN THOUSANDS)
                                                  JUNE 30, 1998
                                                   (UNAUDITED)    DECEMBER 31, 1997
                                                   -----------    -----------------
<S>                                               <C>             <C>
Raw material and supplies.......................    $ 68,953          $ 55,542
Work-in-process and finished goods..............     100,308            95,462
Adjustments to LIFO values......................     (30,272)          (30,272)
                                                    --------          --------
                                                    $138,989          $120,732
                                                    ========          ========
</TABLE>
 
NOTE 6--CREDIT AGREEMENT:
 
     The Company entered into a credit agreement dated June 8, 1998 (the "Credit
Facility"), to replace the Company's existing credit facilities. The unsecured
Credit Facility provides for $125 million in five-year borrowings and $25
million in one-year borrowings, on a revolving basis, of up to the lesser of
$150 million or a borrowing base equal to the sum of 85% of qualified accounts
receivable and 60% of qualified inventory. At June 30, 1998 no amounts were
outstanding under the facility.
 
     Under the terms of the Credit Facility, the Company, at its option, will be
able to borrow at (a) a base rate (which is the higher of PNC Bank's prime rate
or the Federal Funds Effective Rate plus 0.5% per annum), or (b) LIBOR, plus a
spread (ranging from 0.5% to 1.5%) determined by the ratio of the Company's
consolidated total indebtedness to consolidated earnings before interest, taxes,
depreciation and amortization.
 
     The Credit Facility contains additional terms and financial covenants which
are typical for other similar facilities.
 
NOTE 7--SUBSEQUENT EVENTS:
 
     On July 6, 1998, the Company signed a Letter of Intent to acquire Sierra
Alloys Company, Inc. (Sierra), of Irwindale, California, for a combination of
cash and stock. Sierra is a manufacturer of specialty metal products including
stainless and alloy steels, high temperature alloys and titanium. Sierra also
operates as a manufacturing service center providing forged and rolled products
to a number of markets including the aerospace, petrochemical, and medical
implant industries. Sierra's 1997 sales exceeded $50 million.
 
     On July 23, 1998, the Company signed a Letter of Intent to acquire New
Century Metals, Inc. (NCM), of Solon, Ohio. The letter agreement calls for NCM
to exchange all of its outstanding common shares for shares of RMI common stock.
NCM is a manufacturer and distributor of high temperature and corrosion
resistant alloys such as titanium, stainless steel and nickel, in long bar form,
to the aerospace, chemical processing, oil exploration and production, and power
generation industries. In addition to the manufacturing facilities, NCM operates
four distribution centers in the United States and one in England.
 
     These transactions are subject to, among other things, negotiation and
execution of mutually satisfactory definitive agreements, successful completion
of due diligence, the receipt of required regulatory clearances and other
customary pre-closing conditions.
 
ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF RESULTS OF OPERATIONS AND
FINANCIAL CONDITION
 
     The following discussion should be read in connection with the information
contained in the Consolidated Financial Statements and Selected Notes to
Consolidated Financial Statements. The following information contains
"forward-looking statements" within the meaning of the Private Securities
Litigation Reform Act of 1995, and are subject to the safe harbor created by
that Act. Such forward-looking statements include, without limitation,
statements regarding the future availability and prices of raw materials, the
competitiveness of the titanium industry, demand for the Company's products, the
historic cyclicality of the titanium and aerospace industries, uncertain defense
spending, long-term supply agreements, the ultimate determination of pending
trade petitions, global economic conditions, the Company's order backlog and the
conversion of that backlog into revenue and other statements contained herein
that are not historical facts. Because such forward-looking statements involve
risks and uncertainties, there are important factors that could cause actual
results to differ
 
                                        9
<PAGE>   11
 
materially from those expressed or implied by such forward-looking statements.
These and other risk factors are set forth below in the "Outlook" section, as
well as being described in the Company's other filings with the Securities and
Exchange Commission ("SEC") over the last 12 months, copies of which are
available from the SEC or may be obtained upon request from the Company.
 
THREE MONTHS ENDED JUNE 30, 1998 COMPARED TO THREE MONTHS ENDED JUNE 30, 1997
 
NET SALES
 
     Net sales increased to $96.5 million, or 29% for the three months ended
June 30, 1998 compared to net sales of $75.0 million in the corresponding 1997
period. This sales increase is due primarily to increased average selling prices
for titanium mill products, and a shift in product mix to more value added
products such as sheet and plate, and away from commodity type products such as
billet and bloom. Shipments of titanium mill products were 4.9 million pounds in
the second quarter of 1998 compared to 4.6 million pounds for the same period in
1997 despite a three week outage at the Company's rolling mill in April 1998.
Average selling prices on mill products in the second quarter of 1998 increased
by approximately 4% to $14.57 per pound from $14.02 per pound in the second
quarter of 1997. The increase in sales over the second quarter of 1997 is also
partially due to $5.4 million in sales generated by Galt Alloys, Inc. Galt was
acquired in July, 1997.
 
GROSS PROFIT
 
     Gross profit amounted to $24.2 million, or 25.0% of sales for the quarter
ended June 30, 1998 compared to a gross profit of $16.7 million or 22.3% for the
comparable 1997 period. This improvement results primarily from increased profit
margins on titanium mill products resulting from a more favorable product mix
and higher average selling prices.
 
SELLING, GENERAL AND ADMINISTRATIVE EXPENSES
 
     Selling, general and administrative expenses amounted to $3.8 million for
the quarter ended June 30, 1998 compared to $2.8 million for the same quarter in
1997. This increase results primarily from increased business levels. Research,
technical and product development expenses amounted to $1.0 million in the
second quarter of 1998 compared to $0.7 million in the second quarter of 1997.
Selling, general and administrative expenses together with research, technical
and product development expenses amounted to approximately 5.0% of sales in the
second quarter of 1998 compared to 4.7% of sales in the comparable 1997 period.
 
OPERATING INCOME
 
     Operating income for the three months ended June 30, 1998 amounted to $19.4
million, or 20.1% of sales compared to $13.2 million or 17.6% of sales in the
same period of 1997. This improvement results primarily from increased profit
margins on mill products.
 
INCOME TAXES
 
     In the second quarter of 1998, the Company recorded an income tax benefit
of $10.1 million compared to a $1.4 million provision recorded in the second
quarter of 1997. Included in the three month period ended June 30, 1998 is a
$16.1 million benefit resulting from an adjustment to the deferred tax asset
valuation allowance due to changes in the Company's expectations about the
ultimate realization of its deferred tax assets in years after 1998. Exclusive
of this adjustment, the effective federal tax rate for the second quarter of
1998 was approximately 29.0%. The effective federal tax rate for the second
quarter of 1997 was approximately 10.1%. The difference between the statutory
tax rate of 35% and the effective tax rate for both the second quarter of 1998
and 1997 is primarily due to adjustments to the deferred tax asset valuation
allowance as it related to expected current year results. Because of available
net operating loss carry forwards, the amount of current taxes expected to be
paid in 1998 will be well below the level which could be expected by applying
the statutory federal tax rate of 35% to pretax income. See "Income Tax
Considerations" and "SFAS No. 109 Effects" below.
 
                                       10
<PAGE>   12
 
OTHER INCOME
 
     Other income for the second quarter of 1998 amounted to $0.9 million
compared to $0.3 million for the same period in 1997. This increase results
primarily from increased investment income on short term securities.
 
NET INCOME
 
     Net income for the quarter ended June 30, 1998 amounted to $30.4 million or
31.5% of sales compared to $12.0 million, or 16.1% of sales in the comparable
1997 period. This increase is due primarily to increased margins on titanium
mill products and the deferred tax credit referred to above.
 
SIX MONTHS ENDED JUNE 30, 1998 COMPARED TO SIX MONTHS ENDED JUNE 30, 1997
 
NET SALES
 
     Net sales for the six months ended June 30, 1998 increased to $185.6 from
$148.7 million in the first six months of 1997, an increase of $36.8 million, or
24.8%. This increase results primarily from higher average selling prices and
increased mill product shipments. Mill products shipments for the first six
months of 1998 amounted to 9.8 million pounds compared to 9.4 million pounds in
the comparable period of 1997 despite a three week outage at the Company's
rolling mill in April 1998. Average realized selling prices increased to $14.82
per pound in the first six months of 1998 compared to $13.88 per pound in 1997.
The higher average selling prices in the first six months of 1998 compared to
the second quarter is due primarily to shipping less value added flat product as
a result of the rolling mill outage in the second quarter. The increase in sales
from 1997 is also partially due to $10.5 million in sales generated by Galt
Alloys, Inc. Galt was acquired in July, 1997.
 
GROSS PROFIT
 
     Gross profit for the six months ended June 30, 1998 amounted to $49.7
million, or 26.8% of sales compared to $32.5 million, or 21.8% of sales in the
first six months of 1997. This increase results primarily from increased selling
prices for titanium mill products and increased volume.
 
SELLING, GENERAL AND ADMINISTRATIVE EXPENSES
 
     Selling, general and administrative expenses amounted to $7.8 million for
the first six months of 1998 compared to $5.5 million for the comparable period
in 1997. This increase results primarily from increased business levels.
Research, technical and product development expenses amounted to $1.8 million,
and $1.4 million in the six month periods ended June 30, 1998 and 1997,
respectively. Selling, general and administrative expenses together with
research, technical and product development expenses for the first six months of
1998 amounted to 5.2% of sales compared to 4.7% of sales in the first six months
of 1997.
 
OPERATING INCOME
 
     Operating income for the six months ended June 30, 1998 amounted to $40.1
million, or 21.6% of sales compared to $25.6 million, or 17.4% of sales in the
same period of 1997. This improvement results primarily from increases in
average realized selling prices and mill product shipments.
 
INCOME TAXES
 
     In the first six months of 1998, the Company recorded an income tax benefit
of $4.0 million compared to a $2.9 million provision recorded in the first six
months of 1997. The income tax benefit of $4.0 million recorded in the first six
months of 1998 is comprised of an income tax provision of $12.1 million and an
income tax benefit of $16.1 million resulting from an adjustment to reduce the
deferred tax asset valuation allowance due to changes in the Company's
expectations about the ultimate realization of its deferred tax assets.
Exclusive of this adjustment, the effective federal tax rate for the six months
ended June 30, 1998 was approximately 29.0% compared to 11.3% for the same
period in 1997. The difference between the statutory tax rate of 35% and the
effective tax rate for the six month periods ending June 30, 1998 and 1997 is
primarily due
 
                                       11
<PAGE>   13
 
to adjustments to the deferred tax asset valuation allowance related to expected
current year results. Because of available net operating loss carryforwards, the
amount of current taxes expected to be paid in 1998 will be well below the level
which could be expected by applying the statutory federal tax rate of 35% to
pretax income. See "Income Taxes" and "SFAS No. 109 Effects" below.
 
NET INCOME
 
     Net income for the six months ended June 30, 1998 amounted to $45.5
million, or 24.5% of sales compared to $23.0 million, or 15.5% of sales in the
first six months of 1997. This increase is due primarily to increased profit
margins on titanium mill products and the deferred tax credit referred to above.
 
OUTLOOK
 
     During the last three years, and continuing into early 1998, the U.S.
titanium industry has experienced a significant increase in demand and pricing
for titanium mill products. This increase in demand has resulted primarily from
an unprecedented increase in demand from the commercial aerospace markets.
During the 1995-1997 period, most commercial airlines have reported stronger
operating profits. In order to meet increasing passenger load demands and the
need to replace older, less efficient aircraft, commercial airlines placed
orders for new and replacement aircraft at near record levels, and the leading
manufacturers of commercial aircraft announced increased build and delivery
rates for certain aircraft. Recently, Boeing Commercial Airplane Group made
several announcements reducing its forecasted production rates on a number of
wide body aircraft models in the 1999-2000 time frame. Other aerospace
contractors have announced some delivery delays and rescheduling resulting from
the Boeing announcements, and a need to adjust inventory requirements downward
from peak levels. While the company has not experienced significant order
cancellations or rescheduling, pricing on incoming orders has shown some
softening particularly in the nonvalue added commodity products such as ingot
and bloom. Demand and pricing for the Company's core value added products, alloy
sheet and plate, remain relatively strong. This slowing in the growth of
commercial aircraft build rates will impact the overall demand for titanium mill
products. The Company anticipates the balance of 1998 to remain strong both in
terms of mill product shipments and profitability. RMI can give no assurances as
to the extent or duration of demand in the commercial aerospace market.
 
     During the second quarter of 1998, RMI was named by B.F. Goodrich Aerospace
Aerostructures Group, which designs and manufactures engine nacelle systems for
large commercial and military aerospace applications, as its principal supplier
of alloy sheet. During the same period, RMI secured contracts with Contrucciones
Aeronauticas S.A. (CASA) of Spain and Daimler-Benz Aerospace AG of Germany for
their alloy plate and sheet requirements in connection with the Airbus and
Eurofighter programs. All three contracts, totaling more than $60 million, begin
in 1999 and extend for a minimum of three years.
 
     Also, during the second quarter of 1998, RMI was designated the sole
supplier of titanium mill products for the Air Force F-22 fighter being built by
Lockheed Martin and Boeing. The new contract, which begins this year, will
continue through the life of the program with approximately 339 aircraft
forecast to be produced by the year 2012. The value of this contract could
potentially total $340 million.
 
     On February 2, 1998, RMI entered into an agreement with Boeing Commercial
Airplane Group whereby RMI will supply Boeing and its family of commercial
suppliers with up to 4.5 million pounds of titanium products annually. The
agreement, beginning in 1999, will have an initial term of five years and,
subject to review by the parties in the fourth year, could be extended for an
additional five years. Under the accord, Boeing will receive firm prices in
exchange for RMI receiving a minimum volume commitment of 3.25 million pounds
per year, or approximately 50% of the Company's current supply levels to Boeing.
 
     RMI has also been selected by military aircraft producers Boeing and
Northrop as their principal supplier of titanium alloy plate and alloy sheet
including just-in-time, cut-to-size products, for the C-17 Transport, F-15 Eagle
and the F/A-18 Hornet programs. The Hornet program includes the new E/F version
which utilizes considerably more titanium than earlier C/D models. The agreement
will begin with May 1999 requirements and runs through April 2001.
 
                                       12
<PAGE>   14
 
     In another accord, RMI, through its French affiliate, Reamet, has been
chosen by Aerospatiale as the major supplier of the titanium flat rolled
products required for its Airbus programs beginning in 1999 and extending
through 2001. Requirements are principally for flat-rolled products, including
value added cut-to-size shapes.
 
     The Company's total order backlog as of June 30, 1998 was approximately
$368 million, compared to $383 million at December 31, 1997.
 
     Beginning in the second half of 1995 and continuing into 1998, the Company
has experienced a significant increase in the volume of incoming orders at
improving prices. The Company estimates that as of June 30, 1998, orders for
substantially all of its anticipated 1998 shipments have been booked or shipped
at average prices greater than its 1997 average realized mill product selling
price of $14.23 per pound. The Company's average realized mill product selling
price increased to $14.82 per pound in the first six months of 1998 compared to
$13.88 per pound in the first six months of 1997. The Company is currently
booking orders for titanium mill products for delivery in the first half of
1999.
 
     Because of the titanium industry's long-term supply and pricing
arrangements with Boeing and others and the announced changes in aircraft build
rates as discussed above, aerospace customers have become more resistant to
significant price increases. While the Company expects higher average selling
prices for 1998 when compared to the same period in 1997, mill product pricing
is generally beginning to flatten as a result of these long-term arrangements as
well as the slowing of the growth in commercial aircraft build rates.
 
     Prices for the Company's 1998 titanium sponge requirements have been set
under long-term supply contracts and short-term arrangements. In the aggregate,
the Company expects average prices for titanium sponge purchases in 1998 to
approximate 1997 levels. Negotiations covering 1999 sponge requirements and
pricing levels are currently underway. Due to the increase in demand for
titanium mill products, overall demand for titanium scrap, which accounts for
approximately 40% of the Company's raw material requirements has also increased.
However, because of increased supplies of titanium scrap, prices for scrap have
eased somewhat. Prices of certain alloying agents have increased as a result of
increased demand. The Company has announced increased prices and the
implementation of metallic surcharges to recover increased raw material costs,
although there can be no assurances that the Company will continue to be able to
recover these costs.
 
LABOR RELATIONS
 
     United Steelworkers of America ("USWA") represents the production and
maintenance as well as the clerical and technical employees at the Company's
main production facility at Niles, Ohio. The current three year labor agreement
with the USWA expires on September 30, 1998. Negotiations for a new contract
have begun, and the Company is hopeful of reaching a mutually satisfactory
agreement prior to expiration of the current contract.
 
LIQUIDITY AND CAPITAL RESOURCES
 
     Net cash flows from operating activities totaled $21.7 million in the first
six months of 1998 compared to $15.9 million in the first six months of 1997.
The change in net cash flows from operating activities in the first six months
of 1998, compared to the comparable 1997 period was due primarily to
significantly improved results of operations. Working capital amounted to $205.2
million at June 30, 1998, compared to $184.8 million at December 31, 1997. The
increase in working capital results primarily from increases in cash and
inventory. The Company's working capital ratio was 5.3 to 1 at June 30, 1998.
 
     During the first six months of 1998 and the comparable 1997 period, the
Company's cash flow requirements for capital expenditures were funded by cash
provided from operating activities.
 
     The Company anticipates that it will be able to fund its 1998 working
capital requirements and its capital expenditures from funds generated by
operations. The Company's long-term liquidity requirements, including capital
expenditures, are expected to be financed by a combination of internally
generated funds, borrowings and other sources of external financing if needed.
                                       13
<PAGE>   15
 
     Credit Agreement. The Company entered into a credit agreement, dated June
8, 1998 (the "Credit Facility"), to replace the Company's existing credit
facilities. The unsecured Credit Facility provides for $125 million in five-year
borrowings and $25 million in one-year borrowings, on a revolving basis, of up
to the lesser of $150 million or a borrowing base equal to the sum of 85% of
qualifying accounts receivable and 60% of qualifying inventory. At June 30, 1998
no amounts were outstanding under the facility. Under the terms of the Credit
Facility, the Company, at its option will be able to borrow at (a) a base rate
(which is the higher of PNC Bank's prime rate or the Federal Funds Effective
Rate plus 0.5% per annum), or (b) LIBOR plus a spread (ranging from 0.5% to
1.5%) determined by the ratio of the Company's consolidated total indebtedness
to consolidated earnings before interest, taxes, depreciation and amortization.
 
INCOME TAXES
 
     SFAS No. 109 Effects. SFAS 109 requires a valuation allowance when it is
more likely than not that some portion or all of the deferred tax assets will
not be realized. It further states that forming a conclusion that a valuation
allowance is not needed is difficult when there is negative evidence such as
cumulative losses in recent years. The ultimate realization of all or part of
the Company's deferred income tax assets depends on the Company's ability to
generate sufficient taxable income in the future.
 
     When preparing interim and annual financial statements, the Company
periodically evaluates its strategic and business plans, in light of evolving
business conditions, and the valuation allowance is adjusted for future income
expectations resulting from that process, to the extent different from those
inherent in the current valuation allowance. In making an assessment of
realizability at June 30, 1998, the Company considered a number of factors
including the improved profitability in 1998 as a result of increased sales,
product pricing and gross margins, as well as the improved outlook for periods
beyond 1998, when compared to expectations inherent in the December 31, 1997
valuation allowance. Accordingly, the valuation allowance was reduced to reflect
the revised future income expectations. The remaining valuation allowance at
June 30, 1998 relates to deferred tax assets for tax loss carryforwards. These
assets are expected to be realized in 1998. Therefore, pursuant to the
provisions of Accounting Principles Board Opinion No. 28, "Interim Financial
Reporting" and SFAS No. 109, the remaining allowance will be released through
the effective tax rate in the second half of the year.
 
     At December 31, 1997, approximately $16 million of the available net
operating loss carryforwards had not yet been recognized for SFAS 109 purposes.
The valuation allowance relating to net operating loss carryforwards was
recognized in the effective rate calculation for the six months ended June 30,
1998, resulting in an effective book tax rate below the statutory federal rate
of 35%. As a result of this, the entire amount of the Company's net operating
loss carryforwards will be utilized for SFAS 109 purposes in 1998.
 
     For federal income tax return purposes, the amount of remaining net
operating loss carryforwards at December 31, 1997 amounted to approximately $43
million, which are available to offset future taxable income. Because the
Company expects to generate sufficient taxable income, the currently available
net operating loss carryforwards will be fully utilized in 1998. Because of the
availability of these net operating loss carryforwards to offset taxable income,
actual cash payments for federal income taxes in 1998 will be well below the
level which could be expected by applying the statutory federal tax rate of 35%
to pretax income.
 
ENVIRONMENTAL MATTERS
 
     The Company is subject to environmental laws and regulations as well as
various health and safety laws and regulations that are subject to frequent
modifications and revisions. While the costs of compliance for these matters
have not had a material adverse impact on RMI in the past, it is impossible to
predict accurately the ultimate effect these changing laws and regulations may
have on the Company in the future.
 
     At June 30, 1998, the amount accrued for future environment-related costs
was $3.5 million. Based on available information, RMI believes its share of
potential environmental-related costs, before expected contributions from third
parties, is in a range from $4.2 million to $7.0 million, in the aggregate. The
amount accrued is net of expected contributions from third parties (which does
not include any amounts from insurers) of approximately $2.1 million, which the
Company believes are probable. The Company has received
                                       14
<PAGE>   16
 
contributions from such third parties for a number of years as partial
reimbursement for costs incurred by the Company. As these proceedings continue
toward final resolution, amounts in excess of those already provided may be
necessary to discharge the Company from its obligations for these projects.
 
     The ultimate resolution of these environmental matters could individually,
or in the aggregate, be material to the consolidated financial statements.
However, management believes that the Company will remain a viable and
competitive enterprise even though it is possible that these matters could be
resolved unfavorably.
 
CAPITAL EXPENDITURES
 
     Gross capital expenditures the first six months of 1998 and 1997 amounted
to $11.3 and $1.9 million, respectively. The Company has anticipated capital
spending of approximately $30 to $35 million in 1998. RMI anticipates that
current capital spending plans can be funded using cash provided from internally
generated sources.
 
NEW ACCOUNTING STANDARDS
 
     In June 1997, the FASB issued SFAS No. 130, "Reporting Comprehensive
Income". The Company adopted SFAS No. 109 in the first quarter of 1998. Although
this standard requires new reporting for comprehensive income, it does not
impact the measurement of comprehensive income, and, as such, has not had a
material impact on the reported results of operations or financial position.
 
     Also in June 1997, the FASB issued SFAS No. 131, "Disclosures about
Segments of an Enterprise and Related Information." The Company expects to adopt
this standard in its annual financial statements for 1998, but has not yet
completed its evaluation of the impact of this standard. However, since this
standard requires disclosure of additional information but does not prescribe
new measurement standards, the Company does not expect the adoption of this
standard to have a material impact on the reported results of operations or
financial position.
 
     In February 1998, the FASB issued SFAS No. 132, "Employers' Disclosures
about Pensions and Other Postretirement Benefits--an amendment of FASB
Statements No. 87, 88, and 106." This Statement revises employers' disclosures
about pension and other postretirement benefit plans. It does not change the
measurement or recognition of those plans. The Company expects to adopt the
provisions of this standard in its annual financial statements for 1998, and
does not expect this disclosure-only standard to have a significant impact on
reported results of operations or financial position.
 
     In March 1998, the Accounting Standards Executive Committee of the AICPA
issued Statement of Position 98-1, "Accounting for the Costs of Computer
Software Developed or Obtained for Internal Use." The principles established in
this statement are required to be applied to the Company's financial statements
in 1999. Management has not yet completed its evaluation of the impact of this
standard on the Company's reported results of operations or financial position.
 
     In June 1998, the FASB issued SFAS No. 133, "Accounting for Derivative
Instruments and Hedging Activities." This statement establishes standards for
accounting and reporting of derivative financial instruments. The standard is
required to be adopted in 2000. Management has not yet evaluated the impact this
standard will have on reported results of operations or financial position.
 
YEAR 2000 COMPLIANCE
 
     The Company has and will continue to make certain investments in its
application software to ensure the Company is Year 2000 compliant. In addition,
the Company is monitoring the compliance efforts of entities with which it
interacts. The Company began modification of its internally developed
proprietary software in 1996 and currently estimates that approximately 90% of
this software has been modified and tested to make it Year 2000 compliant. The
balance is currently being modified and is expected to be completed in the first
half of 1999. In any event, many of the Company's internally generated software
programs are considered noncritical applications which can, if necessary, be
performed manually. Noninformation technology systems
                                       15
<PAGE>   17
 
such as process control devices and other automated equipment which may contain
imbedded chips that could be affected by the year 2000 problem have been
identified. Modifications to these devices are underway, and the Company
estimates that necessary modifications will be completed in the first quarter of
1999. The Company has also contacted its key vendors, suppliers and service
providers such as financial institutions to monitor their compliance efforts.
While it is not possible at present to accurately estimate the incremental cost
of this effort, the Company does not anticipate it will have a material impact
on the long-term results of operations, liquidity or consolidated financial
position of the Company. This discussion of RMI's efforts and management's
expectations relating to the effect of Year 2000 compliance on operating results
are forward looking statements. RMI's ability to achieve Year 2000 compliance
and the level of incremental costs associated therewith could be adversely
impacted by, among other things, the availability and cost of programming and
testing resources, vendors' ability to modify proprietary software and
unanticipated problems identified in the ongoing compliance review. In addition,
RMI has limited or no control over the actions of proprietary software vendors
and other entities with which it interacts. Therefore, Year 2000 compliance
problems experienced by these entities could adversely affect the results of the
Company.
 
ACQUISITIONS
 
     On July 6, 1998, the Company signed a Letter of Intent to acquire Sierra
Alloys Company, Inc. (Sierra), of Irwindale, California, for a combination of
cash and stock. Sierra is a manufacturer of specialty metal products including
stainless and alloy steels, high temperature alloys and titanium. Sierra also
operates as a manufacturing service center providing forged and rolled products
to a number of markets including the aerospace, petrochemical, and medical
implant industries. Sierra's 1997 sales exceeded $50 million.
 
     On July 23, 1998, the Company signed a Letter of Intent to acquire New
Century Metals, Inc. (NCM), of Solon, Ohio. The letter agreement calls for NCM
to exchange all of its outstanding common shares for shares of RMI common stock.
NCM is a manufacturer and distributor of high temperature and corrosion
resistant alloys such as titanium, stainless steel and nickel, in long bar form
to the aerospace, chemical processing, oil exploration and production, and power
generation industries. In addition to the manufacturing facilities, NCM operates
four distribution centers in the United States and one in England.
 
     These transactions are subject to, among other things, negotiation and
execution of mutually satisfactory definitive agreements, successful completion
of due diligence, the receipt of required regulatory clearances and other
customary pre-closing conditions. Depending on the extent and timing of these
items the Company currently plans on closing the NCM aquisition during the third
quarter of 1998, and the Sierra acquisition in the third or fourth quarter. Any
cash required for these acquisitions will be provided from internally generated
sources, supplemented as required by borrowings under the new credit agreement.
 
                           PART II--OTHER INFORMATION
 
ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K.
 
  (a) Exhibits
 
     3.1 Articles of Incorporation of the Company, as amended June 9, 1997,
         incorporated by reference to Exhibit 10.1 to the Company's Current
         Report on Form 8-K dated June 10, 1997.
 
     3.2 Amended Code of Regulations of the Company, incorporated by reference
         to Exhibit 3.2 to the Company's Annual Report on Form 10-K for the year
         ended December 31, 1993.
 
     4.1 Credit Agreement between RMI Titanium Company and PNC Bank, National
         Association, as agent; National City Bank of Pennsylvania and Bank One,
         National Association as co-agents, dated as of June 8, 1998, filed
         herewith.
 
     27 Financial Data Schedule
 
  (b) There were no reports on Form 8-K filed for the quarter ended June 30,
      1998.
 
                                       16
<PAGE>   18
 
                                   SIGNATURES
 
     Pursuant to the requirements of the Securities Exchange Act of 1934, the
Registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.
 
                                                   RMI TITANIUM COMPANY
                                          --------------------------------------
                                                       (Registrant)
 
Date: August 13, 1998
                                          By:       /s/ T. G. RUPERT
                                              ----------------------------------
                                                        T. G. Rupert
                                             Executive Vice President & Chief
                                                    Financial Officer
 
                                       17