<PAGE>   1
 
================================================================================
 
                       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 MARCH 31, 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)
 
            OHIO                                      31-0875005
(State or other jurisdiction of                    (I.R.S. Employer
incorporation or organization)                    Identification No.)
 
                     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 May 1, 1998, 20,510,752 shares of common stock of the registrant were
outstanding.
 
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<PAGE>   2
 
                              RMI TITANIUM COMPANY
 
                                   FORM 10-Q
                          QUARTER ENDED MARCH 31, 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 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....................   14
Signatures..................................................   15
</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
                                                                  MARCH 31
                                                              -----------------
                                                               1998      1997
                                                               ----      ----
<S>                                                           <C>       <C>
Sales.......................................................  $89,039   $73,708
Operating costs:
Cost of Sales...............................................   63,521    57,932
Selling, general and administrative expenses................    4,006     2,726
Research, technical and product development expenses........      847       704
                                                              -------   -------
     Total operating costs..................................   68,374    61,362
                                                              -------   -------
Operating income............................................   20,665    12,346
Other income-net............................................      542       165
Interest expense............................................       54        33
                                                              -------   -------
Income before income taxes..................................   21,153    12,478
Provision for income taxes (Note 3).........................    6,100     1,560
                                                              -------   -------
Net income..................................................  $15,053   $10,918
                                                              =======   =======
Net income per common share:
Primary.....................................................  $  0.73   $  0.54
                                                              =======   =======
Diluted.....................................................  $  0.73   $  0.52
                                                              =======   =======
</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>
                                                              (UNAUDITED)
                                                               MARCH 31     DECEMBER 31
                                                                 1998          1997
                                                              -----------   -----------
<S>                                                           <C>           <C>
ASSETS
Current assets:
  Cash and cash equivalents.................................   $ 41,059      $ 30,211
  Receivables -- less allowance for doubtful accounts of
     $1,084 and $1,064......................................     69,807        70,898
  Inventories...............................................    128,220       120,732
  Deferred tax asset........................................      1,211         4,811
  Other current assets......................................      5,873         5,903
                                                               --------      --------
     Total current assets...................................    246,170       232,555
  Property, plant and equipment, net of accumulated
     depreciation...........................................     47,059        43,034
  Noncurrent deferred tax asset.............................      5,157         5,157
  Other noncurrent assets...................................     10,430        10,563
                                                               --------      --------
     Total assets...........................................   $308,816      $291,309
                                                               ========      ========
LIABILITIES AND SHAREHOLDERS' EQUITY
Current liabilities:
  Accounts payable..........................................   $ 20,292      $ 25,346
  Accrued wages and other employee costs....................      9,229         8,024
  Other accrued liabilities.................................     20,693        14,361
                                                               --------      --------
     Total current liabilities..............................     50,214        47,731
Long-term debt..............................................         --            --
Accrued postretirement benefit cost.........................     19,376        19,376
Other noncurrent liabilities................................      2,850         3,029
                                                               --------      --------
     Total liabilities......................................     72,440        70,136
                                                               --------      --------
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; 21,088,053 and 21,022,253 shares issued....        210           210
  Additional paid-in capital................................    237,970       236,970
  Retained earnings (deficit)...............................      3,371       (11,682)
  Deferred compensation.....................................     (1,913)       (1,100)
  Treasury Common Stock at cost 577,301 and 575,485
     shares.................................................     (3,262)       (3,225)
                                                               --------      --------
Total shareholders' equity..................................    236,376       221,173
                                                               --------      --------
     Total liabilities and shareholders' equity.............   $308,816      $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>
                                                              THREE MONTHS ENDED
                                                                   MARCH 31
                                                              -------------------
                                                                1998       1997
                                                                ----       ----
<S>                                                           <C>        <C>
CASH PROVIDED FROM (USED IN) OPERATIONS:
Net income..................................................  $15,053    $10,918
Adjustment for items not affecting funds from operations:
  Depreciation..............................................    1,320      1,259
  Deferred taxes............................................    3,600      1,560
  Other-net.................................................      106        122
                                                              -------    -------
                                                               20,079     13,859
                                                              -------    -------
CHANGES IN ASSETS AND LIABILITIES (EXCLUDING CASH):
Receivables.................................................    1,091     (7,482)
Inventories.................................................   (7,488)    (2,068)
Accounts payable............................................   (5,054)     1,943
Other liabilities...........................................    7,358      1,821
Other assets................................................      163        200
                                                              -------    -------
                                                               (3,930)    (5,586)
                                                              -------    -------
     Cash from operating activities.........................   16,149      8,273
                                                              -------    -------
CASH FLOWS FROM INVESTING ACTIVITIES:
  Capital expenditures......................................   (5,345)      (636)
                                                              -------    -------
     Cash used in investing activities......................   (5,345)      (636)
                                                              -------    -------
CASH FLOWS (USED IN) FROM FINANCING ACTIVITIES:
  Net proceeds from issuance of Common Stock................       81        221
  Debt repayments...........................................       --     (2,930)
  Treasury Common Stock purchased...........................      (37)      (147)
                                                              -------    -------
  Cash from (used in) from financing activities.............       44     (2,856)
                                                              -------    -------
INCREASE IN CASH AND CASH EQUIVALENTS.......................   10,848      4,781
CASH AND CASH EQUIVALENTS AT BEGINNING OF PERIOD............  $30,211    $ 5,944
                                                              -------    -------
CASH AND CASH EQUIVALENTS AT END OF PERIOD..................  $41,059    $10,725
                                                              =======    =======
SUPPLEMENTAL CASH FLOW INFORMATION:
  Cash paid for
     Interest (net of amounts capitalized)..................  $    10    $    21
     Income taxes...........................................  $     0    $   130
  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 SHAREHOLDERS' EQUITY
                             (DOLLARS IN THOUSANDS)
<TABLE>
<CAPTION>
                                                                  ADDT'L.    RETAINED    TREASURY        EXCESS
                              SHARES      COMMON     DEFERRED     PAID-IN    EARNINGS     COMMON         MINIMUM
                            OUTSTANDING   STOCK    COMPENSATION   CAPITAL    (DEFICIT)    STOCK     PENSION LIABILITY
                            -----------   ------   ------------   --------   ---------   --------   -----------------
<S>                         <C>           <C>      <C>            <C>        <C>         <C>        <C>
Balance at December 31,
  1996....................  20,290,550     $208      $  (557)     $234,958   $(71,767)   $(3,078)        $(1,028)
Shares issued for
  Directors'
Compensation..............       3,346       --           --            87         --         --              --
Shares issued for
  Restricted Stock Award
  Plans...................      34,950       --         (832)          832         --         --              --
Compensation expense
  recognized..............          --       --          289            --         --         --              --
Treasury Common Stock
  purchased at cost.......      (7,287)      --           --            --         --       (147)             --
Shares issued from
  exercise of employee
  stock options...........     125,209        2           --         1,093         --         --              --
Net income................          --       --           --            --     60,085         --              --
Excess minimum pension
  liability...............          --       --           --            --         --         --           1,028
                            ----------     ----      -------      --------   --------    -------         -------
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..............          --       --          106            --         --         --              --
Treasury Common Stock
  purchased at cost.......      (1,816)      --           --            --         --        (37)             --
Shares issued from
  exercise of employee
  stock options...........      20,675       --           --            81         --         --              --
Net income................          --       --           --            --     15,053         --              --
Excess minimum pension
  liability...............          --       --           --            --         --         --              --
                            ----------     ----      -------      --------   --------    -------         -------
Balance at March 31,
  1998....................  20,510,752     $210      $(1,913)     $237,970   $  3,371    $(3,262)             --
                            ==========     ====      =======      ========   ========    =======         =======
 
<CAPTION>
 
                            COMPREHENSIVE
                               INCOME
                            -------------
<S>                         <C>
Balance at December 31,
  1996....................     $    --
Shares issued for
  Directors'
Compensation..............          --
Shares issued for
  Restricted Stock Award
  Plans...................          --
Compensation expense
  recognized..............          --
Treasury Common Stock
  purchased at cost.......          --
Shares issued from
  exercise of employee
  stock options...........          --
Net income................      60,085
Excess minimum pension
  liability...............       1,028
                               -------
Balance at December 31,
  1997....................      61,113
                               =======
Shares issued for
  Directors'
Compensation..............          --
Shares issued for
  Restricted Stock Award
  Plans...................          --
Compensation expense
  recognized..............          --
Treasury Common Stock
  purchased at cost.......          --
Shares issued from
  exercise of employee
  stock options...........          --
Net income................      15,053
Excess minimum pension
  liability...............          --
                               -------
Balance at March 31,
  1998....................     $15,053
                               =======
</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 the next 12 months.
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 March 31, 1998 approximately 27% of the outstanding
common stock was owned by USX.
 
     In November, 1996, USX Corporation completed a public offering of its notes
exchangeable 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 month periods ended March 31, 1998 and 1997 the Company
recorded an income tax provision of $6.1 million and $1.6 million, respectively.
The effective tax rate for the three months ended March 31, 1998 and 1997 was
approximately 29 percent and 12 percent, respectively. The difference between
the statutory federal tax rate of 35 percent and the effective tax rate is
principally due to adjustments to the deferred tax asset valuation allowance
which existed at December 31 1997 and 1996 as it 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.
 
     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 at this time that a portion of its deferred tax assets will be realized. The
remaining valuation allowance was retained in light of the requirement in SFAS
No. 109 to give weight to objective evidence such as recent losses and the
historical titanium industry business cycle.
 
     When preparing future periods' interim and annual financial statements, the
Company will periodically evaluate its strategic and business plans, in light of
evolving business conditions, and the valuation allowance will be adjusted for
changes in future income expectations resulting from that process.
 
                                        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
Field Brooks 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 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 others 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. It is anticipated that the studies will be completed no
earlier than mid 1998. 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 others 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 March 31, 1998, the amount accrued for future environmental-related
costs was $2.9 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 $3.9 million to $6.7 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)
                                                          MARCH 31, 1998
                                                           (UNAUDITED)     DECEMBER 31, 1997
                                                           -----------     -----------------
<S>                                                       <C>              <C>
Raw material and supplies...............................     $ 62,284          $ 55,542
Work-in-process and finished goods......................       96,208            95,462
Adjustments to LIFO values..............................      (30,272)          (30,272)
                                                             --------          --------
                                                             $128,220          $120,732
                                                             ========          ========
</TABLE>
 
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 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 MARCH 31, 1998 COMPARED TO THREE MONTHS ENDED MARCH 31, 1997
 
NET SALES
 
     Net sales increased to $89.0 million, or 21% for the three months ended
March 31, 1998 compared to net sales of $73.7 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 in the first quarter of
1998 amounted to 4.9 million pounds compared to 4.8 million pounds in the first
quarter of 1997. Average selling prices on mill products in the first quarter of
1998 increased by approximately 10% to $15.06 per pound from $13.75 per pound in
the first quarter of 1997. Pricing on incoming orders for titanium mill products
continues to show improvement.
 
GROSS PROFIT
 
     Gross profit amounted to $25.5 million, or 28.7% of sales for the quarter
ended March 31, 1998 compared to a gross profit of $15.8 million or 21.4% of
sales for the comparable 1997 period. This improvement results primarily from
increased profit margins on titanium mill products resulting from higher average
selling prices and a more favorable product mix.
 
SELLING, GENERAL AND ADMINISTRATIVE EXPENSES
 
     Selling, general and administrative expenses amounted to $4.0 million for
the quarter ended March 31, 1998 compared to $2.7 million for the same quarter
in 1997. Research, technical and product development expenses amounted to $0.8
million in the first quarter of 1998 compared to $0.7 million in the first
quarter of 1997. The increase in selling, general and administrative expenses is
due primarily to increased levels of business activity and the acquisition of
Galt Alloys, Inc. Selling, general and administrative expenses together
 
                                        9
<PAGE>   11
 
with research, technical and product development expenses amounted to
approximately 5.5% of sales in the first quarter of 1998 compared to 4.7% of
sales in the comparable 1997 period.
 
OPERATING INCOME
 
     Operating income for the three months ended March 31, 1998 amounted to
$20.7 million, or 23.2% of sales compared to $12.3 million or 16.7% of sales in
the same period of 1997. This improvement results primarily from significant
increases in profit margins on mill products.
 
INCOME TAXES
 
     In the first quarter of 1998, the Company recorded an income tax provision
of $6.1 million compared to a provision of $1.6 million in the first quarter of
1997. The effective federal tax rate for the first quarter of 1998 was
approximately 29%. The effective tax rate in the first quarter of 1997 was
approximately 12%. The difference between the statutory tax rate of 35% and the
effective tax rate for both the first quarter of 1998 and 1997 is principally
due to adjustments to the deferred tax asset valuation allowance which existed
at the preceding December 31 as it related to expected current year results.
Because of the availability of net operating loss carryforwards to offset
current taxable income, the amount of current taxes expected to be paid in 1998
will be well below the amount 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.
 
NET INCOME
 
     Net income for the quarter ended March 31, 1998 amounted to $15.1 million
or 17% of sales compared to $10.9 million, or 14.8% of sales in the comparable
1997 period.
 
OUTLOOK
 
     During the last three years, and continuing into 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 have been placing
orders for new and replacement aircraft at near record levels, and the leading
manufacturers of commercial aircraft have announced increased build and delivery
rates for certain aircraft. However, changing global or industry economic
conditions could cause the commercial airline industry or the aircraft
manufacturers to re-evaluate aircraft orders, thus affecting aircraft build
rates and the related demand for titanium mill products. RMI can give no
assurances as to the extent or duration of the increased demand in the
commercial aerospace market or the extent to which such recovery will result in
additional demand for titanium products.
 
     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 its current supply levels.
 
     RMI has also been selected by military aircraft producers Boeing and
Northrop as the 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.
 
     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
 
                                       10
<PAGE>   12
 
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 March 31, 1998 was approximately
$388 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 March 31, 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 $15.06 per pound in the first three months of 1998 compared
to $13.75 per pound in the first three months of 1997. The Company is currently
booking orders for titanium mill products for delivery in the first half of
1999. The increase in demand has been driven primarily by increased demand from
the commercial aerospace market.
 
     Assuming a continuing robust aerospace market, RMI expects to continue as a
significant participant in the aerospace markets, and together with its new
market development efforts and expected growth in nonaerosapce application for
titanium believes, but cannot guarantee, it will be able to replace the reduced
Boeing supply community business. However, because of the titanium industry's
long-term supply and pricing arrangements with Boeing and others, aerospace
customers have become more resistant to significant price increases. While
higher average selling prices are anticipated for 1998, the Company does not
anticipate that the rate of pricing improvements in recent years can be
sustained indefinitely.
 
     The increase in demand for titanium products has put upward pressure on
prices for certain raw materials used by the Company. 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 will approximate 1997 levels. 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. Increased scrap generation by
fabricators could increase supplies and thereby put downward pressure on future
scrap prices; however, there is no certainty that supplies will increase or be
offset by other factors. Prices of certain alloying agents have also 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.
 
     The Company anticipates that, due to a three-week outage at its rolling
mill in April, second quarter 1998 shipments of plate and sheet will be reduced
by approximately 400,000 pounds impacting second quarter earnings. The shipments
are expected to be made up in the third and fourth quarters and, therefore, are
not expected to have an impact on 1998 annual earnings.
 
LIQUIDITY AND CAPITAL RESOURCES
 
     Net cash flows from operating activities totaled $16.1 million in the first
three months of 1998 compared to $8.3 million in the first three months of 1997.
The change in net cash flows from operating activities in the first three months
of 1998, compared to the comparable 1997 period was due primarily to
significantly improved results of operations. Working capital amounted to $196.0
million at March 31, 1998, compared to $184.8 million at December 31, 1997. The
increase in working capital results primarily from increases in cash and
inventory partially offset by increases in current liabilities. The Company's
working capital ratio was 4.9 to 1 at March 31, 1998.
 
     During the first three 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.
                                       11
<PAGE>   13
 
     The Company is currently negotiating a new credit agreement to replace its
current $50 million unsecured revolving credit facility which matures April 14,
1999. There are currently no borrowings outstanding under the facility. An
agreement is expected to be in place by the end of the second quarter 1998.
 
INCOME TAX CONSIDERATIONS
 
     SECTION 382 LIMITATION. At December 31, 1997, the Company had net operating
loss carryforwards of approximately $43 million available to reduce federal
taxable income through 2006. If an "ownership change" were to occur, the
utilization of net operating loss carryforwards would be subject to an annual
limitation. Generally, an ownership change occurs with respect to a corporation
if shareholders who own, directly or indirectly, 5% or more of the capital stock
of the corporation increase their aggregate percentage ownership of such stock
by more than 50 percentage points over the lowest percentage of such stock owned
by such shareholders at any time during a prescribed testing period. An
ownership change could result from equity transactions such as exercises of
stock options, purchases or sales of Common Stock by certain stockholders,
including USX and other issuances of Common Stock by the Company. If the annual
limitation were to apply, the amount of the limitation would generally equal the
product of (i) the fair market value of the Company's equity immediately prior
to the ownership change, with certain adjustments, including a possible
adjustment to exclude certain capital contributions made in the two years
preceding the date of the ownership change, and (ii) a long-term tax exempt bond
rate of return published monthly by the Internal Revenue Service. Should the
annual limitation apply, the Company believes that it would not materially
affect the potential use of the net operating loss carryforwards to reduce any
future income tax liabilities over time; however, it is possible that the
Company's results in a particular year could exceed the annual limitation, in
which case such excess would not be reduced by the net operating loss
carryforward and the Company's tax liability would be correspondingly higher.
 
     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 future periods' interim and annual financial statements, the
Company will periodically evaluate its strategic and business plans, in light of
evolving business conditions, and the valuation allowance will be 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 March 31, 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, when compared to expectations inherent in the
December 31, 1997 valuation allowance. Accordingly, the valuation allowance was
adjusted for the difference between such revised future income expectations and
those inherent in the valuation allowance at December 31, 1997.
 
     At December 31, 1997, approximately $16 million of the available net
operating loss carryforwards had not yet been recognized for SFAS 109 purposes.
If the Company were to achieve operating results in 1998 similar to those
achieved in 1997, the remaining valuation allowance relating to net operating
loss carryforwards would be recognized in the effective rate calculation for the
year, resulting in an effective book tax rate below the statutory federal rate
of 35%, but somewhat higher than the 1997 effective rate of 10.3%. Should this
occur, 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 purpose, 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. If the Company
were to generate sufficient taxable income, the currently available net
operating loss carryforwards could be fully utilized in 1998. Because of the
availability of these net operating loss carryforwards to offset taxable income,
the Company estimates that 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.
 
                                       12
<PAGE>   14
 
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 March 31, 1998, the amount accrued for future environment-related costs
was $2.9 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 $3.9 million to $6.7 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 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 three months of 1998 and 1997 amounted
to $5.3 and $0.6 million, respectively. The Company has anticipated capital
spending of approximately $35 to $40 million in 1998. RMI anticipates that
current capital spending plans can be funded using cash provided from internally
generated sources. New Accounting Standards
 
NEW ACCOUNTING STANDARDS
 
     In June 1997, the FASB issued SFAS No. 130, "Reporting Comprehensive
Income," and SFAS No. 131, "Disclosures about Segments of an Enterprise and
Related Information." SFAS No. 130 establishes standards for reporting and
display of comprehensive income and its components. The Company adopted the
provisions of SFAS No. 130 in its consolidated financial statements for the
three months ending March 31, 1998. SFAS No. 131 requires certain disclosures
about segment information in interim and annual financial statements and related
information about products and services, geographic areas and major customers.
The Company must adopt the provisions of SFAS No. 131 for its consolidated
financial statements for the year ending December 31, 1998.
 
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 estimates that approximately 90% of its internally
developed proprietary software has been modified to make it Year 2000 compliant.
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.
 
                                       13
<PAGE>   15
 
                          PART II -- OTHER INFORMATION
 
ITEM 4. SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS.
 
     The annual meeting of stockholders was held on April 24, 1998. In
connection with the meeting, proxies were solicited pursuant to the Securities
Exchange Act. The following are the voting results on proposals considered and
voted upon at the meeting, all of which were described in the proxy statement.
 
     1. All nominees for directors listed in the proxy statement were elected.
        Listed below are the names of each director elected, together with their
        individual vote totals.
 
<TABLE>
<CAPTION>
                                                                 VOTES         VOTES
                                                                  FOR        WITHHELD
                                                               ----------    ---------
        <S>                                                    <C>           <C>
        Craig R. Andersson...................................  16,274,540    2,183,269
        Neil A. Armstrong....................................  16,273,933    2,183,876
        Daniel I. Booker.....................................  16,299,653    2,158,174
        Ronald L. Gallatin...................................  16,298,719    2,159,090
        Charles C. Gedeon....................................  16,272,610    2,185,199
        Robert M. Hernandez..................................  16,067,088    2,390,721
        Dana J. Johnson......................................  16,288,828    2,168,981
        John H. Odle.........................................  16,275,385    2,182,424
        Timothy G. Rupert....................................  16,274,585    2,183,224
        Wesley W. von Schack.................................  16,298,933    2,158,876
</TABLE>
 
     2. Price Waterhouse LLP was elected as independent accountants for 1998:
        For, 16,544,987; against 17,597; abstained; 1,895,225.
 
     3. Proposal to amend the Company's Articles of Incorporation to increase
        the authorized number of Common Shares from 30,000,000 to 80,000,000. A
        2/3 majority of all outstanding shares eligible to vote was required to
        amend the Articles of Incorporation. 12,669,653 shares (62% of the
        outstanding shares) voted for and 5,100,028 shares (25% of the
        outstanding shares) voted against the proposal. 2,710,000 shares
        abstained from voting or were not voted. The required 2/3 majority was
        not obtained and the proposal was not approved.
 
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.
 
     10.1 Employment agreement, dated March 6, 1998 between the Company and
          Lawrence W. Jacobs, filed herewith.
 
      27 Financial Data Schedule
 
  (b) There were no reports on Form 8-K filed for the quarter ended March 31,
1998.
 
                                       14
<PAGE>   16
 
                                   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: May 8, 1998
                                          By:       /s/ T. G. RUPERT
 
                                            ------------------------------------
                                                        T. G. Rupert
                                             Executive Vice President & Chief
                                                    Financial Officer
 
                                       15