Exhibit 5.1

[LETTERHEAD OF FULBRIGHT & JAWORSKI L.L.P.]

July 21, 2004

Alexion Pharmaceuticals, Inc.

352 Knotter Drive

Cheshire, CT 06410

                Re:    Alexion Pharmaceuticals, Inc (the “Company”)

Dear Sirs:

Reference is made to the Underwriting Agreement, dated July 20, 2004 (the “Underwriting Agreement”), by and among the Company and Morgan Stanley & Co. Incorporated and SG Cowen & Co., LLC, as underwriters (the “Underwriters”). We have acted as counsel to the Company in connection with (i) the purchase by the Underwriters of an aggregate of 5,000,000 shares (the “Firm Shares”) of the Company’s common stock, $.0001 par value per share (the “Common Stock”), and (ii) the grant to the Underwriters of an option to purchase up to an additional 500,000 shares (the “Additional Shares”) of Common Stock to cover over-allotments, if any, in the sale of the Firm Shares by the Underwriters. The Firm Shares and the Additional Shares are collectively referred to herein as the “Shares.” A prospectus supplement, dated July 20, 2004, has been filed under Rule 424(b)(5) of the Securities Act of 1933, as amended (the “Act”), on July 21, 2004, with respect to the sale of the Shares, to the prospectus dated May 14, 2004 which was part of a Registration Statement on Form S-3 (Registration No. 333-114449), as amended, which became effective under the Act on May 14, 2004.

We have examined such corporate records, other documents and questions of law as we have deemed necessary or appropriate for the purposes of this opinion.

Upon the basis of such examination, we advise you that, in our opinion, the Shares have been duly and validly authorized and, when issued and paid for in accordance with the Underwriting Agreement, will be legally issued, fully paid and non-assessable.

We consent to the filing of this opinion as an exhibit to a Current Report on Form 8-K which is incorporated by reference into the Registration Statement and to the reference to this firm under the caption “Legal Matters” in the Supplement contained therein and elsewhere in the Registration Statement and Supplement. This consent is not to be construed as an admission that we are a party whose consent is required to be filed with the Registration Statement under the provisions of the Act.

Very truly yours,

/s/    Fulbright & Jaworski L.L.P.