<PAGE>   1



                                   FORM 10-Q


                                 UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                            Washington, D.C.  20549

                QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d)
                     OF THE SECURITIES EXCHANGE ACT OF 1934

                    For the quarter ended September 30, 1998

                          Commission File No. 0-10810

                              KIEWIT ROYALTY TRUST
             (Exact name of Registrant as specified in its charter)


<TABLE>
<S>                                                   <C>
           Nebraska                                          47-6131402
-------------------------------                       --------------------------------------------
(State or other jurisdiction of                             (I.R.S. Employer Identification No.)
 incorporation or organization)

</TABLE>

                                 Trust Division
                         U.S. Bank National Association
                               1700 Farnam Street
                               Omaha, Nebraska  68102
                      ------------------------------------
                    (Address of Principal Executive Offices)

                                  (402) 348-6000
                  --------------------------------------------
                (Registrant's telephone no. 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
                                                               ---      ---

                    APPLICABLE ONLY TO CORPORATE REGISTRANTS

                 Indicate the number of shares outstanding of each of the
registrant's classes of common stock, as of the latest practicable date.

                 As of September 30, 1998, there were 12,633,432 Units of
Beneficial Interest of the registrant outstanding.
<PAGE>   2
                         PART I - FINANCIAL INFORMATION

ITEM 1.   FINANCIAL STATEMENTS.

                              KIEWIT ROYALTY TRUST
               STATEMENT OF ASSETS, LIABILITIES AND TRUST CORPUS

                 as of September 30, 1998 and December 31, 1997

<TABLE>
<CAPTION>
ASSETS                                                   1998                1997
------                                                   ----                ----

<S>                                                   <C>                 <C>
Cash equivalents                                      $2,897,099          $ 180,832

Royalty and overriding
royalty interests in coal
leases                                                $  167,817          $ 167,817

Less accumulated amortization                            (96,048)           (85,611)
                                                      -----------         ----------

Net royalty and overriding
royalty interests in coal leases                      $   71,769          $  82,206
                                                      -----------         ----------

Total Assets                                          $2,968,868          $ 263,038
                                                      ===========         ==========

LIABILITIES AND TRUST CORPUS
----------------------------

Distribution payable to unit holders                  $2,897,099          $ 180,832

Trust corpus: 12,633,432 units of beneficial
interest authorized and outstanding                   $   71,769          $  82,206
                                                      -----------         ----------

Total Liabilities and Trust Corpus                    $2,968,868          $ 263,038
                                                      ===========         ==========

</TABLE>

                  The accompanying notes are an integral part
                          of the financial statements





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<PAGE>   3
                              KIEWIT ROYALTY TRUST


                       STATEMENT OF DISTRIBUTABLE INCOME

     For the Three Months and Nine Months Ended September 30, 1998 and 1997


<TABLE>
<CAPTION>
                                     Three Months Ended                      Nine Months Ended
                                        September 30                            September 30
                                ----------------------------           ------------------------------

                                 1998                   1997            1998                1997
                                 ----                   ----            ----                ----

<S>                          <C>                   <C>                 <C>                <C>
Royalty income               $2,884,143            $3,229,866          $5,622,543         $5,318,310
                    
Interest income                  23,975                27,739              49,798             49,577

Trust expenses                  (11,021)              (10,433)            (41,776)           (37,962)
                                --------              --------            --------           --------

Distributable income         $2,897,099            $3,247,172          $5,630,565         $5,329,925
                    
Distributable income
per unit (12,633,432          $.2293201             $.2570301           $.4456877          $.4218905
Units Outstanding)              
</TABLE>



                      STATEMENT OF CHANGES IN TRUST CORPUS

             For the Nine Months Ended September 30, 1998 and 1997


<TABLE>
<CAPTION>


                                                         1998                  1997
                                                        -------              --------

             <S>                                        <C>                  <C>
                                              
             Trust corpus as of January 1            $     82,206          $     90,896


             Amortization of royalty interests            (10,437)               (4,867)


             Distributable income                       2,733,466             5,329,925


             Distribution to unit holders              (2,733,466)           (5,329,925)
                                                       -----------          ------------

             Trust corpus as of September 30         $     71,769          $     86,029
                                                      ===========           ===========

</TABLE>

                  The accompanying notes are an integral part
                          of the financial statements





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<PAGE>   4
                              KIEWIT ROYALTY TRUST

                         Notes to Financial Statements
                 as of September 30, 1998 and December 31, 1997

BASIS OF PRESENTATION

                 The accompanying unaudited financial statements have been
prepared in accordance with the instructions for Form 10-Q and do not
necessarily include all of the information and footnotes required by generally
accepted accounting principles for complete financial statements.  In the
Trustee's opinion, all adjustments necessary for a fair presentation have been
included.  For further information, refer to the financial statements and
footnotes included in the Trust's annual report on Form 10-K for the year ended
December 31, 1997.

SUMMARY OF SIGNIFICANT ACCOUNTING POLICIES

                 The accompanying unaudited financial statements have been
                 prepared on the following basis:

                 (a)        The royalty income recorded for a month is the
                            amount computed and paid by the lease holder to the
                            Trustee for such month.

                 (b)        Trust administration expenses are recorded in the
                            month they accrue.

This basis for reporting royalty income is thought to be the most meaningful
because distributions to the Unit Holders for a month are based on net cash
receipts for such month.  However, these statements differ from financial
statements prepared in accordance with generally accepted accounting principles
because, under such principles, royalty income for a month would be based on
production for such month without regard to when calculated or received.  In
addition, amortization of the net royalty and overriding royalty interests,
which is calculated on a units-of-production basis by lease, is charged
directly to trust corpus, since such amount does not affect distributable
income.





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<PAGE>   5
ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS
OF OPERATIONS.

                 a.         MATERIAL CHANGES IN FINANCIAL CONDITION.

                            Kiewit Royalty Trust is a royalty trust with
royalty and overriding royalty interests in certain coal leases.  The Trust was
formed for the purposes of (1) administering the income received from such coal
leases and (2) distributing such income (together with interest earned thereon
less payment of or provision for obligations) to the holders of the Units of
Beneficial Interest.

                 b.         MATERIAL CHANGES IN RESULTS OF OPERATIONS.

                            During the nine months ended September 30, 1998,
the Trust received a total of $5,622,543 of royalty and overriding royalty
payments; during the three months ended September 30, it received a total of
$2,884,143 of such payments.  The following schedule reflects the royalty and
overriding royalty amounts received by the Trust from each mine.

<TABLE>
<CAPTION>
                                        Three Months Ended                 Nine Months Ended
           Mine                            September 30                      September 30
           ----                    ------------------------------     ------------------------------

                                        1998           1997               1998           1997
                                        ----           ----               ----           ----
           <S>                        <C>           <C>                 <C>          <C>
           Decker                     $1,902,589     $2,346,807         $4,531,489     $4,420,919

           Spring Creek                  974,358        872,941            974,358        872,941


           Black Butte                         -             -                  -            (274)

           Big Horn                        7,196         10,118            116,696         24,724
                                        --------      ---------            -------      ---------

                                      $2,884,143     $3,229,866         $5,622,543     $5,318,310
                                      ==========     ==========         ==========     ==========

</TABLE>
                 i.     DECKER MINE.

                 The amount of royalties and overriding royalties received by
the Trust with respect to the Decker Mine increased to $4,531,489 in the nine
months ended September 30, 1998 compared to $4,420,919 during the same period
in 1997.  For the third quarter of 1998, the amounts received decreased to
$1,902,589 compared to $2,346,807 during the third quarter of 1997.  These
variations in receipts during the nine months and three months ended September
30, 1998 were the net result due to (i) changes in the relative amounts of coal
mined under leases bearing high and low overriding royalty rates per ton, which
are a normal result of the execution of a mining plan encompassing several coal
leases bearing different





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<PAGE>   6
royalty rates and (ii) a lower average selling price of coal mined resulting
from the expiration of a long-term coal contract which was replaced by several
other contracts.

             ii.     SPRING CREEK MINE.

                 Royalties received from the Spring Creek Mine increased to
$974,358 in the nine months ended September 30, 1998 compared to $872,941
during the same period in 1997.  The Trust received $974,358 in the third
quarter of 1998 compared to $872,941 during the same period in 1997 from the
applicable lease.  Royalties from the Spring Creek Mine are typically paid by
the mine operators on an annual basis during the second half of the calendar
year.


              iii.     BLACK BUTTE MINE.

                 The Trust received no royalties during the first nine months
of 1998, as compared to the payment of $274 made by the Trust to Black Butte
Mine during the first nine months of 1997.  The Trust received no royalties
during the three months ended September 30, 1998, as compared to the payment of
$274 made by the Trust to the Black Butte Mine during the same period in 1997.
The reason no royalties are being received is that the trust no longer receives
any overriding royalties on coal produced after March 31, 1996 (as explained
below) as a result of the renewal of Federal Lease Number W-6266 effective
April 1, 1996.

                 Lease number W-6266 provides that the terms and conditions of
the lease are subject to "reasonable readjustment" effective on April 1, 1996.
At that time, the royalty rate payable to the United States was increased to 12
1/2%, which is the currently prevailing federal royalty rate.  Because of the
cap on the amount of the overriding royalty payable to the Trust with respect
to this lease, the increased royalty rate has effectively eliminated further
payments to the Trust with respect to this lease.





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<PAGE>   7
             iv.     BIG HORN MINE.

                 Overall royalties received from the Big Horn Mine increased to
$116,696 during the nine months ended September 30, 1998 from $24,724 in 1997,
and decreased to $7,196 in the third quarter of 1998 from $10,118 in the third
quarter of 1997.  The increase in the first nine months of 1998 reflects
increased production from the applicable leases.

                 c.                 YEAR 2000 ISSUE.

                  The "Year 2000" computer problem has arisen because many
computer applications worldwide will not properly recognize the date change
from December 31, 1999, to January 1, 2000, potentially causing production of
erroneous data, miscalculations, system failures and other operational
problems.

                 Since the Trust does not have any computer system of its own,
initiatives to address the Year 2000 issue has been undertaken by the Trustee,
U.S. Bank National Association, to evaluate the Year 2000 impact on its
critical computer hardware and software.  The Trust does not incur any cost in
the Trustee's addressing of its Year 2000 problem.  The Trustee has indicated
to the Trust that it has developed a strategic plan focusing on achieving Year
2000 compliance. Certain systems are being replaced, modified or reprogrammed
to be Year 2000 compliant.  If necessary, new hardware and software is being
purchased to be Year 2000 compliant.  Systems are being tested in an isolated
environment dedicated to Year 2000 testing.  The Year 2000 project includes
contingency plans to mitigate potential delays or other problems, such as
back-up solutions for all mission-critical applications and business
continuation plans for significant vendors and other business partners.  The
Year 2000 efforts of the Trustee and any related third parties are not within
the Trust's control, however, and their failure to remediate Year 2000 issues
successfully could result in business disruption, increased operating costs and
increased credit risk for the Trust.  At the present time, it is not possible
to determine whether any such events are likely to occur, or to quantify any
potential negative impact they may have on the Trust's future results of
operations and financial condition.

                 Ultimately, the potential impact of the Year 2000 issue will
depend not only on the success of the corrective measures undertaken by the
Trustee, but also on the way in which the Year 2000 issue is addressed by
vendors, service providers, counterparties, utilities, governmental agencies
and other entities with which the Trustee and the Trust do





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<PAGE>   8
business.  These parties are receiving communication from the Trustee to
heighten their awareness of the Year 2000 issue, to learn how the Trustee is
addressing it and to evaluate any likely impact on the Trustee and the Trust.

                 The foregoing discussion regarding Year 2000 contains
forward-looking statements, which are based on management's best estimates
derived using various assumptions.  These forward-looking statements involve
inherent risks and uncertainties, and actual results could differ materially
from those contemplated by such statements. Such material differences could
result in, among other things, business disruption, operational problems,
financial loss, legal liability and similar risks.

                          PART II - OTHER INFORMATION.

ITEM 1. LEGAL PROCEEDINGS.

        There are no material pending legal proceedings to which the Trust is a
party or which any of its property is the subject.

ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K.

        (a)      Exhibits

        4.1      Kiewit Royalty Trust Indenture dated May 17, 1982, as amended
                 June 9, 1982 and June 23, 1982 (filed as Exhibit 1 to the
                 Trust's Form 10 filed with the Securities and Exchange
                 Commission on December 23, 1982, and incorporated herein by
                 reference).

        4.2      Order dated September 23, 1994, of the County Court of Douglas
                 County, Nebraska (filed as Exhibit 2 to the Trust's Form 10-Q
                 filed with the Securities and Exchange Commission on November
                 14, 1994, and incorporated herein by reference).

        27.1     Financial Data Schedule.

        (b)      No reports on Form 8-K were filed during the quarter for which
                 this report is filed.





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                                   SIGNATURE

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.

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<S>                    <C>
                       KIEWIT ROYALTY TRUST
                       (Registrant)
                 
                       By:  U. S. Bank National Association
                       in its capacity as
                       Trustee and not in its individual capacity or otherwise



                       By:  /s/ Susan K. Rosburg
                            ----------------------
                            Susan K. Rosburg
                            Trust Officer
</TABLE>


Dated:  November 11, 1998





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