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                       SECURITIES AND EXCHANGE COMMISSION

                             Washington, D.C. 20549

                                    FORM 8-K

                                 CURRENT REPORT

     Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934

                                 May 29, 1998
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                Date of Report (Date of earliest event reported)

                             COSTILLA ENERGY, INC.
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            (Exact name of registrant as specified in its character)

     Delaware                          0-21411                   75-2658940
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 (State or other jurisdiction)   (Commission File Number)     (IRS  Employer 
                                                             Identification No.)

                        400 West Illinois, Suite 1000
                            Midland, Texas 79701
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                  (Address of principal executive offices)

                               (915) 683-3902
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            (Registrant's telephone number, including area code)

                                     N/A
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        (Former name or former address, if changed since last report)





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ITEM 5.  OTHER EVENTS

         The exchange offer (the "Exchange Offer") of Costilla Energy, Inc.
(the "Registrant"), whereby the Registrant offered to exchange registered 10
1/4% Senior Notes due 2006 (the "Exchange Notes") for $80.0 million of
outstanding unregistered 10 1/4% Senior Notes due 2006 (the "Private Notes"),
which was commenced on April 28, 1998, expired at 5:00 p.m., Eastern time, on
May 29, 1998.  The Exchange Offer was made by the Registrant pursuant to that
certain Registration Rights Agreement (the "Registration Rights Agreement")
between the Registrant and BT Alex. Brown Incorporated and Prudential
Securities Incorporated, the initial purchasers of the Private Notes. $77.0
million of the $80.0 million of Private Notes were surrendered in exchange for
Exchange Notes.  The Exchange Notes were registered under the Securities Act of
1933, as amended, by the filing of a registration statement on Form S-4 that
became effective on April 28, 1998, and are freely tradeable subject to certain
limitations described in the Prospectus relating to the Exchange Offer.  The
remaining $3.0 million of Private Notes will remain outstanding and are
unregistered and subject to applicable restrictions on transfer.  With only
certain limited exceptions, the Exchange Offer satisfied the Registrant's
obligations under the Registration Rights Agreement.

         The Exchange Notes are issued under that certain Indenture dated as of
October 1, 1996, as amended by that certain First Supplemental Indenture dated
as of January 16, 1998, between the Registrant and State Street Bank and Trust
Company, as Trustee (the "Indenture").  The remaining Private Notes and an
outstanding issue of $100.0 million of 10 1/4% Senior Notes are also issued
under the Indenture.  A copy of the Indenture has been included as an exhibit
to the Registrant's filings with the Securities and Exchange Commission (see
Item 7 hereof).

ITEM 7. FINANCIAL STATEMENTS AND EXHIBITS

         (a) and (b)      No financial statements are required

         (c)              Exhibits (Incorporated by reference to the filings
                          indicated)

                          4.1     Indenture dated as of October 1, 1996 by and
                                  between State Street Bank and Trust Company,
                                  as Trustee, and Costilla Energy, Inc., as
                                  Issuer, filed as an exhibit to the
                                  Registrant's Registration Statement on Form
                                  S-1, File No. 333-08909.

                          4.2     Form of Notes or Global Certificate (included
                                  as Exhibit A to Exhibit 4.1).

                          4.3     First Supplemental Indenture dated as of
                                  January 16, 1998 by and between State Street
                                  Bank and Trust Company, as Trustee, and
                                  Costilla Energy, Inc., as Issuer (including
                                  form of Global Certificate), filed as an
                                  exhibit to the Registrant's Annual Report on
                                  Form 10-K for the year ended December 31,
                                  1997.



                                   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 hereunto duly authorized.

                                        COSTILLA ENERGY, INC.


                                        By: \s\ BOBBY W. PAGE                   
      Dated: June 12, 1998                  -----------------------------------
                                            Bobby W. Page, Senior
                                            Vice President and
                                            Chief Financial Officer