Exhibit 32

CERTIFICATION OF CHIEF EXECUTIVE OFFICER

I, Charles R. Williamson, Chairman of the Board, Chief Executive Officer and President 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 Annual Report on Form 10-K of the Company for the period ended December 31, 2003 (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: March 11, 2004

/s/ CHARLES R. WILLIAMSON

Charles R. Williamson

CERTIFICATION OF CHIEF FINANCIAL OFFICER

I, Terry G. Dallas, Executive Vice President and 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 Annual Report on Form 10-K of the Company for the period ended December 31, 2003 (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: March 11, 2004

/s/ TERRY G. DALLAS

Terry G. Dallas