================================================================================
                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                                   FORM 10-Q

[X]  QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE
     SECURITIES EXCHANGE ACT OF 1934 FOR THE QUARTERLY PERIOD 
     ENDED SEPTEMBER 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 1-8432

                               MESA OFFSHORE TRUST
             (EXACT NAME OF REGISTRANT AS SPECIFIED IN ITS CHARTER)

                    TEXAS                                76-6004065
       (STATE OR OTHER JURISDICTION OF                (I.R.S. EMPLOYER
       INCORPORATION OR ORGANIZATION)                IDENTIFICATION NO.)

            CHASE BANK OF TEXAS,
            NATIONAL ASSOCIATION
          CORPORATE TRUST DIVISION
               712 MAIN STREET
               HOUSTON, TEXAS                               77002
  (ADDRESS OF PRINCIPAL EXECUTIVE OFFICES)               (ZIP CODE)

                                 (713) 216-6369
              (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 [ ]

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

     As of November 10, 1998 -- 71,980,216 Units of Beneficial Interest in Mesa
Offshore Trust.

================================================================================
<PAGE>
                        PART I -- FINANCIAL INFORMATION

ITEM 1.  FINANCIAL STATEMENTS

                              MESA OFFSHORE TRUST

                       STATEMENTS OF DISTRIBUTABLE INCOME
                                  (UNAUDITED)

<TABLE>
<CAPTION>
                                          THREE MONTHS ENDED         NINE MONTHS ENDED
                                            SEPTEMBER 30,              SEPTEMBER 30,
                                       ------------------------  --------------------------
                                          1998         1997          1998          1997
                                       ----------  ------------  ------------  ------------
<S>                                    <C>         <C>           <C>           <C>         
Royalty income.......................  $  535,215  $  1,756,412  $  1,670,489  $  4,654,552
Interest income......................      42,747        37,287        98,328        90,700
General and administrative expense...     (74,688)     (220,704)     (281,678)     (870,770)
                                       ----------  ------------  ------------  ------------
     Distributable income............  $  503,274  $  1,572,995  $  1,487,139  $  3,874,482
                                       ==========  ============  ============  ============
     Distributable income per unit...  $    .0070  $      .0219  $      .0207  $      .0538
                                       ==========  ============  ============  ============
</TABLE>

               STATEMENTS OF ASSETS, LIABILITIES AND TRUST CORPUS

                                        SEPTEMBER 30,      DECEMBER 31,
                                             1998              1997
                                       ----------------  ----------------
                                         (UNAUDITED)

               ASSETS
Cash and short-term investments......  $      2,460,527  $      2,993,764
Interest receivable..................            42,747            32,568
Net overriding royalty interest in
  oil and gas properties.............       380,905,000       380,905,000
Accumulated amortization.............      (380,847,972)     (380,656,800)
                                       ----------------  ----------------
                                       $      2,560,302  $      3,274,532
                                       ================  ================
    LIABILITIES AND TRUST CORPUS
Reserve for Trust expenses...........  $      2,000,000  $      2,000,000
Distributions payable................           503,274         1,026,332
Trust corpus (71,980,216 units of
  beneficial interest authorized and 
  outstanding).......................            57,028           248,200
                                       ----------------  ----------------
                                       $      2,560,302  $      3,274,532
                                       ================  ================

  (The accompanying notes are an integral part of these financial statements.)

                                       1
<PAGE>
                              MESA OFFSHORE TRUST

                     STATEMENTS OF CHANGES IN TRUST CORPUS
                                  (UNAUDITED)
<TABLE>
<CAPTION>
                                            THREE MONTHS ENDED             NINE MONTHS ENDED
                                              SEPTEMBER 30,                  SEPTEMBER 30,
                                       ----------------------------  -----------------------------
                                           1998           1997           1998            1997
                                       ------------  --------------  -------------  --------------
<S>                                    <C>           <C>             <C>            <C>           
Trust corpus, beginning of period....  $    118,279  $      917,002  $     248,200  $    1,062,405
     Distributable income............       503,274       1,572,995      1,487,139       3,874,482
     Distributions to unitholders....      (503,274)     (1,572,995)    (1,487,139)     (3,874,482)
     Amortization of net overriding
       royalty interest..............       (61,251)        (88,113)      (191,172)       (233,516)
                                       ------------  --------------  -------------  --------------
Trust corpus, end of period..........  $     57,028  $      828,889  $      57,028  $      828,889
                                       ============  ==============  =============  ==============
</TABLE>

  (The accompanying notes are an integral part of these financial statements.)

                                       2
<PAGE>
                              MESA OFFSHORE TRUST
                         NOTES TO FINANCIAL STATEMENTS
                                  (UNAUDITED)

NOTE 1 -- TRUST ORGANIZATION

     The Mesa Offshore Trust (the "Trust") was created effective December 1,
1982 when Mesa Petroleum Co., predecessor to Mesa Limited Partnership, which was
the predecessor to MESA Inc., transferred a 99.99% interest in the Mesa Offshore
Royalty Partnership (the "Partnership") to the Trust. The Partnership was
created to receive and hold a 90% net overriding royalty interest (the
"Royalty") in ten producing and nonproducing oil and gas properties located in
federal waters offshore Louisiana and Texas (the "Royalty Properties"). Until
August 7, 1997, MESA Inc. owned and operated its assets through Mesa Operating
Co. ("Mesa"), the operator and the managing general partner of the Royalty
Properties. On August 7, 1997, MESA Inc. merged with and into Pioneer Natural
Resources Company ("Pioneer") formerly a wholly owned subsidiary of MESA, Inc.
and Parker & Parsley Petroleum Company merged with and into Pioneer Natural
Resources USA, Inc. (successor to Mesa Operating Co.), a wholly owned subsidiary
of Pioneer ("PNR") (collectively, the mergers are referred to herein as the
"Merger"). Subsequent to the Merger, Pioneer owns and operates its assets
through PNR and is also the managing general partner of the Partnership. As used
in this report, the term PNR generally refers to the operator of the Royalty
Properties, unless otherwise indicated.

STATUS OF THE TRUST

     In 1996, PNR drilled five wells from the existing "A" platform on the
South Marsh Island 155 block. PNR recovered all remaining costs related to the
South Marsh Island drilling program as of the February 1997 reporting month. In
addition, during the first quarter of 1997, the Trust recovered approximately
$.5 million in general and administrative expenses paid from the Trust's reserve
fund during the period in which Royalty income was not paid to the Trust,
replenishing the Trust's expense reserve fund balance to $2 million.

     The Trust Indenture provides that the Trust will terminate if the total
amount of cash per year received by the Trust falls below certain levels for
each of three successive years. The December 31, 1997 reserve report prepared
for the Partnership (see the Trust's 1997 Annual Report on Form 10-K) indicates
that 95% of future net revenues will be received by the Trust during the next
four years. As such, it is possible, depending on the timing of future
production, market conditions, the success of future drilling activities, if
any, and other matters, that in 1998 the Trust may commence a period of three
successive years in which annual net royalty income would be below the
termination threshold prescribed in the Indenture, resulting in termination of
the Trust pursuant to the terms discussed above.

NOTE 2 -- BASIS OF PRESENTATION

     The accompanying unaudited financial information has been prepared by Chase
Bank of Texas, National Association (the "Trustee") in accordance with the
instructions to Form 10-Q, and the Trustee believes such information includes
all the disclosures necessary to make the information presented not misleading.
The information furnished reflects all adjustments which are, in the opinion of
the Trustee, necessary for a fair presentation of the results for the interim
periods presented. The financial information should be read in conjunction with
the financial statements and notes thereto included in the Trust's 1997 Annual
Report on Form 10-K.

                                       3
<PAGE>
     The financial statements of the Trust are prepared on the following basis:

          (a)  Royalty income recorded for a month is the Trust's interest in
     the amount computed and paid by PNR to the Partnership for such month
     rather than either the value of a portion of the oil and gas produced by
     PNR for such month or the amount subsequently determined to be 90% of the
     net proceeds for such month;

          (b)  Interest income, interest receivable and distributions payable to
     unitholders include interest to be earned on short-term investments from
     the financial statement date through the next distribution date;

          (c)  Trust general and administrative expenses are recorded in the
     month they accrue;

          (d)  Amortization of the net overriding royalty interest, which is
     calculated on the basis of current royalty income in relation to estimated
     future royalty income, is charged directly to trust corpus since such
     amount does not affect distributable income; and

          (e)  Distributions payable are determined on a monthly basis and are
     payable to unitholders of record as of the last business day of each month.
     However, cash distributions are made quarterly in January, April, July and
     October, and include interest earned from the monthly record dates to the
     date of distribution.

     This basis for reporting royalty income is considered to be the most
meaningful because distributions to the unitholders for a month are based on net
cash receipts for such month. However, it will differ from the basis used for
financial statements prepared in accordance with generally accepted accounting
principles in several respects. Under such principles, royalty income for a
month would be based on net proceeds from production for such month without
regard to when calculated or received and interest income would be calculated
only for the periods covered by the financial statements and would exclude
interest from the period end to the date of distribution.

     The instruments conveying the Royalty provide that PNR will calculate and
pay the Partnership each month an amount equal to 90% of the net proceeds for
the preceding month. Generally, net proceeds means the excess of the amounts
received by PNR from sales of oil and gas from the Royalty Properties plus other
cash receipts over operating and capital costs incurred.

                                       4
<PAGE>
ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS 
        OF OPERATIONS

NOTE REGARDING FORWARD-LOOKING STATEMENTS

     This Form 10-Q includes "forward-looking statements" within the meaning
of Section 27A of the Securities Act of 1933, as amended, and Section 21E of the
Securities Exchange Act of 1934, as amended. All statements other than
statements of historical facts included in this Form 10-Q, including without
limitation the statements under "Management's Discussion and Analysis of
Financial Condition and Results of Operations" and Note 1 to the financial
statements of the Trust regarding the future net revenues of the Trust, are
forward-looking statements. Although Pioneer has advised the Trust that it
believes that the expectations reflected in such forward-looking statements are
reasonable, no assurance can be given that such expectations will prove to have
been correct. Important factors that could cause actual results to differ
materially from expectations ("Cautionary Statements") are disclosed in this
Form 10-Q, including without limitation in conjunction with the forward-looking
statements included in this Form 10-Q. All subsequent written and oral
forward-looking statements attributable to the Trust or persons acting on its
behalf are expressly qualified in their entirety by the Cautionary Statements.

INFORMATION SYSTEMS FOR THE YEAR 2000

     Pioneer has established a "Year 2000" project (the "Project") to assess, to
the extent possible, its Year 2000 risk exposure; to take remedial actions
necessary to minimize the Year 2000 risk exposure to Pioneer and third parties
with whom it has data interchange; and, to test its systems and processes once
remedial actions have been taken. Because of the importance of occurrence dates
in the oil and gas industry, the consequences of not pursuing Year 2000
modifications could be critical to Pioneer's ability to manage and report
operating activities. Pioneer has contracted with a third party to perform the
assessment and remedial phases of its Year 2000 project.

     The assessment phase of the Project is 85% complete as of September 30,
1998, and included, among other procedures, the assessment of information
technology applications and systems, the assessment of non-information
technology exposures; the initiation of inquiry and dialogue with significant
third party business partners, customers and suppliers in an effort to
understand their Year 2000 readiness and potential impact on Pioneer; and the
formulation of contingency plans for mission-critical information technology
systems. Pioneer expects to complete the assessment phase of its Year 2000
project by the end of the first quarter of 1999 but is being delayed by limited
responses to inquiries made of third party businesses. The remedial phase of the
Project is approximately 40% complete as of September 30, 1998, subject to the
results of third party inquiry assessments and the testing phase. The
remediation of non-information technology is expected to be completed by
mid-1999. The testing phase of the Project is expected to be completed by May
1999 for information technology systems and by mid-1999 for non-information
technology.

     Although Pioneer is making every effort to mitigate the risks associated
with the Year 2000 problem, there can be no assurance that the Project or
resulting contingency plans will have anticipated all Year 2000 scenarios. A
failure to remedy a critical Year 2000 problem could result in information and
non-information technology failures, the receipt or transmission of erroneous
data, lost data or a combination of similar problems of a magnitude that cannot
be accurately assessed at this time.

     Chase Bank of Texas, National Association ("Chase" or the "Trustee")
has developed and is implementing a program to prepare its systems and
applications for the Year 2000, including those used to render services to the
Trust. In that connection, Chase intends to have such systems and applications
capable of processing, on and after January 1, 2000, date, and date-related data
consistent with the

                                       5
<PAGE>
functionality of such systems and applications, without a material adverse
effect upon its performance of services as Trustee.

FINANCIAL REVIEW

     During the third quarter of 1998, the Trust had distributable income of
$503,274, representing $.0070 per unit, as compared to $1,572,995, representing
$.0219 per unit in the third quarter of 1997. The per unit amounts of
distributable income for the third quarter of 1998 and 1997 were earned by month
as follows:

                                         1998       1997
                                       ---------  ---------
July.................................  $   .0010  $   .0074
August...............................      .0019      .0080
September............................      .0041      .0065
                                       ---------  ---------
                                       $   .0070  $   .0219
                                       =========  =========

     Royalty income decreased to $535,215 in the third quarter of 1998 as
compared to Royalty income for the third quarter of 1997. The decrease in
Royalty income is primarily due to both reduced production and lower average
sales prices for crude oil, condensate and natural gas liquids in the third
quarter of 1998 when compared to the corresponding 1997 period.

     Production volumes for natural gas decreased to 359,001 Mcf in the third
quarter of 1998 from 798,555 Mcf in the third quarter of 1997.The average price
received for natural gas was $2.21 per Mcf in the third quarter of 1998 compared
to $2.12 per Mcf in the third quarter of 1997.

     Crude oil, condensate and natural gas liquids production decreased to
25,153 barrels in the third quarter of 1998 from 40,870 barrels in the third
quarter of 1997. The average price received for crude oil, condensate and
natural gas liquids was $12.00 per barrel in the third quarter of 1998, compared
to $14.05 per barrel in the third quarter of 1997.

     The decrease in natural gas and crude oil, condensate and natural gas
liquids production for both the nine months and the quarter ended September 30,
1998 when compared to the comparable periods of 1997 are primarily attributable
to the cessation of production on the A-6 ST and A-21 wells on South Marsh
Island blocks 155 and 156 during late 1997 and natural production declines on
the A-20, A-22 and A-14 ST wells as a result of pressure depletion and water
encroachment.

     For the nine months ended September 30, 1998, natural gas production
volumes decreased to 1,164,666 Mcf from 3,491,449 Mcf for the nine months ended
September 30, 1997. Crude oil, condensate and natural gas liquids production
volumes decreased to 49,593 barrels in the first nine months of 1998 as compared
to 147,274 barrels in the first nine months of 1997.

OPERATIONAL REVIEW

     PNR has advised the Trust that during the third quarter of 1998 its
offshore gas production was marketed under short term contracts at spot market
prices to multiple purchasers, including Columbia Energy Service and Enron Gas
Marketing, and that it expects to continue to market its production under short
term contracts for the foreseeable future. Spot market prices for natural gas in
the third quarter of 1998 were generally lower than spot market prices in the
third quarter of 1997.

     The amount of cash distributed by the Trust is dependent on, among other
things, the sales prices and quantities of gas, crude oil, condensate and
natural gas liquids produced from the Royalty Properties and the quantities
sold. Substantial uncertainties exist with regard to future gas and oil prices,
which are subject to fluctuations due to the regional supply and demand for
natural gas and oil, production levels and other

                                       6
<PAGE>
activities of OPEC and other oil and gas producers, weather, storage levels,
industrial growth, conservation measures, competition and other variables.

     The Brazos A-39 block experienced a decrease in natural gas production in
the third quarter of 1998 as compared to the third quarter of 1997, primarily
due to natural production decline. The Brazos A-7 block also experienced a
decrease in natural gas production in the third quarter of 1998 compared to the
third quarter of 1997, primarily due to natural production decline. PNR farmed
out a portion of the Brazos A-7 block to another operator and participated at a
10% working interest in the completion of an exploratory gas well that was
drilled in the second quarter of 1997. The No. 5 well encountered gas pay and
was suspended pending completion operations. Production facilities were
installed, and the No. 5 well commenced production late in the second quarter at
a rate of approximately 10 MMcf per day. The combined completion and facility
costs are expected to total $6.6 million ($594,000 net to the Trust).

     The South Marsh Island 155 and 156 blocks experienced a decrease in
production in the third quarter of 1998 as compared to the third quarter 1997,
primarily due to natural production decline, and the cessation of production in
the A6 ST. Production for the block is currently 1.6 MMcf per day and 74 barrels
of condensate per day as of November 1998. PNR purchased 3D seismic data for the
South Marsh Island 156 block at a cost of $300,000 ($189,000 net to the Trust).
The data has been evaluated and PNR has no current plans for additional
drilling.

     The West Delta 61 and 62 blocks experienced a decrease in production in the
third quarter of 1998 as compared to the third quarter of 1997, primarily due to
the cessation of production of PNR operated wells. These wells are currently
uneconomic to produce. In portions of West Delta Block 62, the Trust is
receiving royalty income from this property pursuant to a farmout agreement with
another operator. The interest in the farmout wells which is attributable to the
Trust consists of a 7.5% overriding royalty interest. In West Delta Block 61,
PNR farmed out portions of the block to another operator, retaining a 10% (9%
net to the Trust) overriding royalty interest. A new well was drilled in the
second quarter which encountered 320 net feet of pay in 8 Miocene sands below a
true vertical depth of 7,500 feet. The operator will drill two development wells
beginning later this year or in early 1999. The Trust will receive an 11.25%
overriding royalty interest in these wells.

     Matagorda Island 624 production decreased in the third quarter of 1998 as
compared to the third quarter of 1997, primarily due to natural production
decline. Gross production for the block is currently 2.2 MMcf per day and 32
barrels of condensate per day as of November 1998.

                             PENDING SALE AGREEMENT

     In September 1998, PNR signed a purchase and sale agreement (the
"Agreement") to sell certain oil and gas properties. Included with the
properties to be sold are Trust properties. The sale is expected to close by
December 31, 1998 and, at such time, the purchaser will become the operator of
the properties. The consummation of this Agreement is primarily dependent upon
the purchaser's successful ability to finance the purchase as well as certain
other contingencies defined in the Agreement. As a result, there can be no
assurance that the sale of any or all of these properties will be completed. In
addition, if such a sale is consummated, the Trust has been advised that the
sale should have no significant effects on the Trust, although the precise
nature of any effects cannot be predicted or quantified at this time.

TERMINATION OF THE TRUST

     The terms of the Mesa Offshore Trust Indenture provide that the Trust will
terminate upon the first to occur of the following events: (1) the total amount
of cash received per year by the Trust for each of three successive years
commencing after December 31, 1987 is less than 10 times one-third of the total
amount payable to the Trustee as compensation for such three year period or (2)
a vote by the unitholders in favor of termination. Because the Trust will
terminate in the event the total amount of cash received per year by the Trust
falls below certain levels, it would be possible for the Trust to terminate even
though some of the

                                       7
<PAGE>
Royalty Properties continued to have remaining productive lives. For information
regarding the estimated remaining life of each of the Royalty Properties and the
estimated future net revenues of the Trust based on information provided by PNR,
see the Trust's 1997 Annual Report on Form 10-K. Upon termination of the Trust,
the Trustee will sell for cash all the assets held in the Trust estate and make
a final distribution to unitholders of any funds remaining after all Trust
liabilities have been satisfied. The discussion set forth above is qualified in
its entirety by reference to the Trust Indenture itself, which is available upon
request from the Trustee. Amounts paid to the Trustee as compensation were
$173,000, $123,000 and $149,000 for the years 1997, 1996, and 1995,
respectively.

     The December 31, 1997, reserve report prepared for the Partnership (see the
Trust's 1997 Annual Report on Form 10-K) indicates that 95% of future net
revenues will be received by the Trust during the next four years. As such, it
is possible, depending on the timing of future production, market conditions,
the success of future drilling activities, if any, and other matters, that in
1998 the Trust may commence a period of three successive years in which annual
net royalty income would be below the termination threshold prescribed in the
Indenture, resulting in termination of the Trust pursuant to the terms discussed
above.

     The terms of the First Amended and Restated Articles of General Partnership
of the Partnership provide that the Partnership shall dissolve upon the
occurrence of any of the following: (a) December 31, 2030; (b) the election of
the Trustee to dissolve the Partnership; (c) the termination of the Trust; (d)
the bankruptcy of the Managing General Partner; or (e) the dissolution of the
Managing General Partner or its election to dissolve the Partnership; provided
that the Managing General Partner shall not elect to dissolve the Partnership so
long as the Trustee remains the only other partner of the Partnership. In the
event of a dissolution of the Partnership and a subsequent winding up and
termination thereof, the assets of the Partnership (i.e., the Royalty interest)
could either (i) be distributed in kind ratably to the Managing General Partner
and the Trustee or (ii) be sold and the proceeds thereof distributed ratably to
the Managing General Partner and the Trustee. In the event of a sale of the
Royalty and a distribution of the cash proceeds to the Trustee, the Trustee
would make a final distribution to unitholders of such cash proceeds plus any
other cash held by the Trust after the payment of or provision for all
liabilities of the Trust, and the Trust would be terminated.

                                       8
<PAGE>
     The following tables provide summaries of the calculations of the net
proceeds attributable to the Partnership's royalty interests (unaudited):

<TABLE>
<CAPTION>
                                                     SOUTH
                                        BRAZOS       MARSH        WEST      MATAGORDA
                                        A-7 AND   ISLAND 155    DELTA 61      ISLAND
                                         A-39       AND 156      AND 62        624          TOTAL
                                       ---------  -----------  ----------   ----------   -----------
<S>                                    <C>        <C>          <C>           <C>         <C>        
THREE MONTHS ENDED
  SEPTEMBER 30, 1998:
    Ninety percent of gross
      proceeds.......................  $ 242,529  $   697,366  $   (2,509)   $158,565    $ 1,095,951
    Less ninety percent of --
      Operating expenditures.........    (78,729)    (281,960)   (144,102)    (35,891)      (540,682)
      Capital costs recovered........     --          --           --          --            --
      Accrual for future abandonment
         costs.......................    (11,727)      (1,500)     (5,848)       (925)       (20,000)
                                       ---------  -----------  ----------   ----------   -----------
    Net proceeds (excess costs)......  $ 152,073  $   413,906  $ (152,459)   $121,749    $   535,269
                                       =========  ===========  ==========   ==========   ===========
    Trust share of net proceeds
      (99.99%).......................                                                    $   535,215
                                                                                         ===========
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................        127       23,391      --           1,635         25,153
                                       =========  ===========  ==========   ==========   ===========
      Average sales price per Bbl....  $    9.53  $     11.11  $   --        $  24.87    $     12.00
                                       =========  ===========  ==========   ==========   ===========
      Natural gas (Mcf)..............    110,908      186,887      (2,613)     63,819        359,001
                                       =========  ===========  ==========   ==========   ===========
      Average sales price per Mcf....  $    2.18  $      2.34  $      .96    $   1.85    $      2.21
                                       =========  ===========  ==========   ==========   ===========
    Producing wells..................          3            3           3           1             10

                                                     SOUTH
                                        BRAZOS       MARSH        WEST      MATAGORDA
                                        A-7 AND   ISLAND 155    DELTA 61      ISLAND
                                         A-39       AND 156      AND 62        624          TOTAL
                                       ---------  -----------  ----------   ----------   -----------
THREE MONTHS ENDED
  SEPTEMBER 30, 1997:
    Ninety percent of gross
      proceeds.......................  $ 340,092  $ 1,231,930  $  467,587    $229,683    $ 2,269,292
    Less ninety percent of --
      Operating expenditures.........   (104,913)    (271,174)   (204,273)    (52,818)      (633,178)
      Capital costs recovered........     --          174,948     (26,909)     --            148,039
      Accrual for future abandonment
         costs.......................    (20,608)       3,610     (10,426)       (141)       (27,565)
                                       ---------  -----------  ----------   ----------   -----------
    Net proceeds (excess costs)......  $ 214,571  $ 1,139,314  $  225,979    $176,724    $ 1,756,588
                                       =========  ===========  ==========   ==========   ===========
    Trust share of net proceeds
      (99.99%).......................                                                    $ 1,756,412
                                                                                         ===========
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................        313       29,433       9,038       2,086         40,870
                                       =========  ===========  ==========   ==========   ===========
      Average sales price per Bbl....  $   16.88  $     12.48  $    18.11    $  18.17    $     14.05
                                       =========  ===========  ==========   ==========   ===========
      Natural gas (Mcf)..............    159,498      405,857     137,420      95,780        798,555
                                       =========  ===========  ==========   ==========   ===========
      Average sales price per Mcf....  $    2.10  $      2.13  $     2.21    $   2.00    $      2.12
                                       =========  ===========  ==========   ==========   ===========
    Producing wells..................          3            4           3           2             12
</TABLE>

------------
o   The amounts shown are for Mesa Offshore Royalty Partnership.

o   The amounts for the three months ended September 30, 1998 and 1997 represent
    actual production for the periods May 1998 through July 1998 and May 1997
    through July 1997, respectively.

o   Capital costs recovered represent capital costs incurred during the current
    or prior periods to the extent that such costs have been recovered by PNR
    from current period Gross Proceeds.

o   Producing wells indicate the number of wells capable of production as of the
    end of the period.

o   West Delta 61 and 62 have ceased production since the second quarter of
    1998. However, operating expenses are still being incurred for maintenance
    procedures.

o   The average sales price per Bbl for Matagorda Island 624 is impacted by
    prior period adjustments received during the third quarter of 1998.

                                       9
<PAGE>

<TABLE>
<CAPTION>
                                                       SOUTH
                                         BRAZOS        MARSH         WEST      MATAGORDA
                                         A-7 AND     ISLAND 155    DELTA 61     ISLAND
                                          A-39        AND 156       AND 62        624         TOTAL
                                        ---------    ----------   ----------   ---------   -----------
<S>                                     <C>          <C>          <C>          <C>         <C>        
NINE MONTHS ENDED
  SEPTEMBER 30, 1998:
    Ninety percent of gross
      proceeds.......................   $ 800,332    $1,422,381   $  452,700   $ 720,753   $ 3,396,166
    Less ninety percent of --
      Operating expenditures.........    (236,107)     (789,506)    (500,112)   (137,902)   (1,663,627)
      Capital costs recovered........      --            --           --          (1,883)       (1,883)
      Accrual for future abandonment
         costs.......................     (35,181)       (4,499)     (17,544)     (2,776)      (60,000)
                                        ---------    ----------   ----------   ---------   -----------
    Net proceeds (excess costs)......   $ 529,044    $  628,376   $  (64,956)  $ 578,192   $ 1,670,656
                                        =========    ==========   ==========   =========   ===========
    Trust share of net proceeds
      (99.99%).......................                                                      $ 1,670,489
                                                                                           ===========
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................         560        43,643          594       4,796        49,593
                                        =========    ==========   ==========   =========   ===========
      Average sales price per Bbl....   $   12.76    $    12.53   $    15.51   $   18.24   $     13.12
                                        =========    ==========   ==========   =========   ===========
      Natural gas (Mcf)..............     344,684       369,828      170,106     280,048     1,164,666
                                        =========    ==========   ==========   =========   ===========
      Average sales price per Mcf....   $    2.30    $     2.37   $     2.61   $    2.26   $      2.36
                                        =========    ==========   ==========   =========   ===========
    Producing wells..................           3             3            3           1            10

                                                       SOUTH
                                         BRAZOS        MARSH         WEST       MATAGORDA
                                         A-7 AND     ISLAND 155    DELTA 61      ISLAND
                                          A-39        AND 156       AND 62         624         TOTAL
                                        ---------    ----------   -----------   ---------   ------------
NINE MONTHS ENDED
  SEPTEMBER 30, 1997:
    Ninety percent of gross
      proceeds.......................   $1,382,672   $7,953,873   $ 1,450,559   $1,011,701  $ 11,798,805
    Less ninety percent of --
      Operating expenditures.........    (313,451)   (1,002,961)     (615,845)   (190,447)    (2,122,704)
      Capital costs recovered........      --        (4,773,208)      (60,806)     (4,564)    (4,838,578)
      Accrual for future abandonment
        costs........................    (105,724)      (15,128)      (52,939)     (8,714)      (182,505)
                                        ---------    ----------   -----------   ---------   ------------
    Net proceeds (excess costs)......   $ 963,497    $2,162,576   $   720,969   $ 807,976   $  4,655,018
                                        =========    ==========   ===========   =========   ============
    Trust share of net proceeds
      (99.99%).......................                                                       $  4,654,552
                                                                                            ============
    Production Volumes and Average
      Prices:
      Crude oil, condensate and
         natural gas liquids
         (Bbls)......................         864       127,523        11,542       7,345        147,274
                                        =========    ==========   ===========   =========   ============
      Average sales price per Bbl....   $   18.80    $    16.66   $     18.30   $   20.08   $      16.97
                                        =========    ==========   ===========   =========   ============
      Natural gas (Mcf)..............     542,701     2,141,543       475,597     331,608      3,491,449
                                        =========    ==========   ===========   =========   ============
      Average sales price per Mcf....   $    2.52    $     2.72   $      2.61   $    2.61   $       2.66
                                        =========    ==========   ===========   =========   ============
    Producing wells..................           3             4             3           2             12
</TABLE>

------------
o   The amounts shown are for Mesa Offshore Royalty Partnership.

o   The amounts for the nine months ended September 30, 1998 and 1997 represent
    actual production for the periods November 1997 through July 1998, and
    November 1996 through July 1997, respectively.

o   Capital costs recovered represent capital costs incurred during the current
    or prior periods to the extent that such costs have been recovered by PNR
    from current period Gross Proceeds.

o   Producing wells indicate the number of wells capable of production as of the
    end of the period.

o   West Delta 61 and 62 have ceased production since the second quarter of
    1998. However, operating expenses are still being incurred for maintenance
    procedures.

o   The average sales price per Bbl for Matagorda Island 624 is impacted by
    prior period adjustments received during the third quarter of 1998.

                                       10
<PAGE>
                                    PART II

ITEM 6.  EXHIBIT AND REPORTS ON FORM 8-K

     (A)  EXHIBITS

     (Asterisk indicates exhibit previously filed with the Securities and
Exchange Commission and incorporated herein by reference.)

<TABLE>
<CAPTION>
                                                                                                 SEC FILE
                                                                                                    OR
                                                                                               REGISTRATION    EXHIBIT
                                                                                                  NUMBER       NUMBER
                                                                                               ------------    -------
<S>                 <C>                                                                           <C>             <C>   
       4(a)        *Mesa Offshore Trust Indenture between Mesa Petroleum Co. and Texas
                    Commerce Bank National Association, as Trustee, dated December 15,
                    1982....................................................................      2-79673         10(gg)
       4(b)        *Overriding Royalty Conveyance between Mesa Petroleum Co. and Mesa
                    Offshore Royalty Partnership, dated December 15, 1982...................      2-79673         10(hh)
       4(c)        *Partnership Agreement between Mesa Offshore Management Co. and Texas
                    Commerce Bank National Association, as Trustee, dated December 15,
                    1982....................................................................      2-79673         10(ii)
       4(d)        *Amendment to Partnership Agreement between Mesa Offshore Management Co.,
                    Texas Commerce Bank National Association, as Trustee, and Mesa Operating
                    Limited Partnership, dated December 27, 1985 (Exhibit 4(d) to Form 10-K
                    for year ended December 31, 1992 of Mesa Offshore Trust)................       1-8432          4(d)
       4(e)        *Amendment to Partnership Agreement between Texas Commerce Bank National
                    Association, as Trustee and Mesa Operating Limited Partnership dated as
                    of January 5, 1994 (Exhibit 4(e) to Form 10-K for year ended December
                    31, 1993 of Mesa Offshore Trust)........................................       1-8432          4(e)
      27            Financial Data Schedule
</TABLE>

     (B)  REPORTS ON FORM 8-K

          None.

                                       11
<PAGE>
                                   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.

                                          MESA OFFSHORE TRUST

                                          By CHASE BANK OF TEXAS,
                                             NATIONAL ASSOCIATION
                                                  TRUSTEE

                                          By /s/ PETE FOSTER
                                                 PETE FOSTER
                                                 SENIOR VICE PRESIDENT 
                                                 & TRUST OFFICER

Date:  November 12, 1998

     The Registrant, Mesa Offshore Trust, has no principal executive officer,
principal financial officer, board of directors or persons performing similar
functions. Accordingly, no additional signatures are available and none have
been provided.

                                       12