<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 June 30, 1998

                           Commission File No. 0-10810

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


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


                                 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 June 30, 1998, there were 12,633,432 Units of Beneficial Interest
of the registrant outstanding.




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                         PART I - FINANCIAL INFORMATION

ITEM 1.     FINANCIAL STATEMENTS.

                              KIEWIT ROYALTY TRUST
                 STATEMENT OF ASSETS, LIABILITY AND TRUST CORPUS

                    as of June 30, 1998 and December 31, 1997

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

<S>                                            <C>                   <C>        
Cash and short-term investments                  $  197,923            $  180,832

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

Less accumulated amortization                    $  (92,583)           $  (85,611)
                                                 ----------            ----------

Net royalty and overriding
royalty interests in coal leases                 $   75,234            $   82,206
                                                 ----------            ----------

Total Assets                                     $  273,157            $  263,038
                                                 ==========            ==========


LIABILITIES AND TRUST CORPUS

Distributions payable to Unit Holders            $  197,923            $  180,832

Trust Corpus:  12,633,432
Units of Beneficial Interest
authorized and outstanding                       $   75,234            $   82,206
                                                 ----------            ----------

Total Liabilities and Trust Corpus               $  273,157            $  263,038
                                                 ==========            ==========
</TABLE>



                   The accompanying notes are an integral part
                           of the financial statements




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                              KIEWIT ROYALTY TRUST


                        STATEMENT OF DISTRIBUTABLE INCOME

        For the Three Months and Six Months Ended June 30, 1998 and 1997


<TABLE>
<CAPTION>
                                     Three Months Ended                 Six Months Ended
                                          June 30                           June 30
                                 --------------------------        ---------------------------

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

<S>                              <C>              <C>              <C>              <C>       
Royalty Income                   $ 203,430        $ 238,339        $2,738,400       $2,088,444

Interest Income                      6,867            5,496            25,822           21,838

Trust Expenses                     (12,374)         (12,641)         ( 30,756)         (27,529)
                                 ---------        ---------        ----------       ----------

Distributable
  Income                         $ 197,923        $ 231,194        $2,733,466       $2,082,753

Distributable
  income per unit
  (12,633,432 units)             $.0156666        $.0183002        $ .2163677       $ .1648604
</TABLE>



                      STATEMENT OF CHANGES IN TRUST CORPUS

                 For the Six Months Ended June 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               (6,972)            (1,239)

Distributable Income                         2,733,466          2,082,753

Distribution to Unit Holders                (2,733,466)        (2,082,753)
                                             ---------        -----------

Trust Corpus as of June 30                   $  75,234        $    89,657
                                             =========        ===========
</TABLE>




    The accompanying notes are an integral part of the financial statements




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                              KIEWIT ROYALTY TRUST

                          Notes to Financial Statements
                    as of June 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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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 six months ended June 30, 1998, the Trust
received a total of $2,738,400 of royalty and overriding royalty payments;
during the three months ended June 30, it received a total of $203,430 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             Six Months Ended
  Mine                         June 30                          June 30
 ------                  ------------------------      --------------------------
                           1998          1997             1998            1997
                         --------     -----------      ----------     -----------
<S>                      <C>          <C>              <C>            <C>         
Black Butte              $    ---     $      (274)     $      ---     $      (274)
Decker                    158,230         237,064       2,628,900       2,074,112
Big Horn                   45,200           1,549         109,500          14,606
Spring Creek                  ---             ---             ---            - --
                         --------     -----------      ----------     -----------
                         $203,430     $   238,339      $2,738,400     $ 2,088,444
                         ========     ===========      ==========     ===========
</TABLE>


           i.     BLACK BUTTE MINE.

         The Trust received no royalties during the first six months of 1998, as
compared to the payment of $274 made by the Trust to Black Butte Mine during the
first six months of 1997. The Trust received no royalties during the three
months ended June 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 




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<PAGE>   6

with respect to this lease, the increased federal royalty rate has effectively
eliminated further payments to the Trust with respect to this lease.

           ii.     DECKER MINE.

         The amount of royalties and overriding royalties received by the Trust
with respect to the Decker Mine increased to $2,628,900 in the first six months
of 1998 compared to $2,074,112 during the same period in 1997. For the second
quarter, the amounts received decreased to $158,230 compared to the $237,064
received during the second quarter of 1997. These changes in royalty and
overriding royalty amounts received were due to 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 royalty rates.

           iii.     BIG HORN MINE.

         Royalties received from the Big Horn Mine increased to $45,200 during
the second quarter of 1998 from $1,549 in 1997, and overall receipts for the
first six months of 1998 increased to $109,500 compared to $14,606 during the
corresponding period in 1997. These increases for 1998 reflect increased
production in the relative amounts of coal mined under leases bearing high and
low overriding royalty rates per ton.

           iv.      SPRING CREEK MINE. 

         No royalties were received from the Spring Creek Mine during the first
six months of 1998 or 1997 because royalties are typically paid by the mine
operators on an annual basis during the second half of a calendar year.




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                          PART II - OTHER INFORMATION.

ITEM 1.  LEGAL PROCEEDINGS. 

         There are no material pending legal proceedings to which the Trust is a
party of 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-Q 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.

                              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



Dated:  August 3, 1998




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