Exhibit 32

                    CERTIFICATION OF CHIEF EXECUTIVE OFFICER


I, Charles R. Williamson, Chief Executive Officer of Unocal Corporation (the
"Company"), certify, pursuant to Section 906 of the Sarbanes-Oxley Act of 2002,
18 U.S.C. Section 1350, that, to my knowledge:

(1)      the Quarterly Report on Form 10-Q of the Company for the quarterly
         period ended March 31, 2004 (the "Report"), fully complies with the
         requirements of Section 13(a) or 15(d) of the Securities Exchange Act
         of 1934 (15 U.S.C. 78m or 78o(d)); and

(2)      the information contained in the Report fairly presents, in all
         material respects, the financial condition and results of operations of
         the Company.


Dated:  May 7, 2004

                                                /s/ CHARLES R. WILLIAMSON
                                                --------------------------------
                                                    Charles R. Williamson



                    CERTIFICATION OF CHIEF FINANCIAL OFFICER

I, Terry G. Dallas, Chief Financial Officer of Unocal Corporation (the
"Company"), certify, pursuant to Section 906 of the Sarbanes-Oxley Act of 2002,
18 U.S.C. Section 1350, that to my knowledge:

(1)      the Quarterly Report on Form 10-Q of the Company for the quarterly
         period ended March 31, 2004 (the "Report"), fully complies with the
         requirements of Section 13(a) or 15(d) of the Securities Exchange Act
         of 1934 (15 U.S.C. 78m or 78o(d)); and

(2)      the information contained in the Report fairly presents, in all
         material respects, the financial condition and results of operations of
         the Company.


Dated:  May 7, 2004

                                                /s/ TERRY G. DALLAS
                                                ------------------------------
                                                    Terry G. Dallas