Exhibit 5.1

+1 212 230 8800 (t)

+1 212 230 8888 (f)

wilmerhale.com

July 31, 2018

State Street Corporation

One Lincoln Street

Boston, Massachusetts 02111

Re: Prospectus Supplement to Registration Statement on Form S-3

Ladies and Gentlemen:

This opinion is furnished to you in connection with (i) the Registration Statement on Form S-3 (File No. 333-221293) (the “Registration Statement”) filed by State Street Corporation, a Massachusetts corporation (the “Company”), with the Securities and Exchange Commission (the “Commission”) on November 2, 2017, for the purposes of registering with the Commission under the Securities Act of 1933, as amended (the “Securities Act”) certain securities of the Company, including shares of the Company’s common stock, par value $1.00 per share (the “Common Stock”), which may be issued from time to time on a delayed or continuous basis pursuant to Rule 415 under the Securities Act at an initial aggregate offering price not to exceed $6,000,000,000, as set forth in the Registration Statement and the prospectus contained therein (the “Base Prospectus”), and (ii) the prospectus supplement, dated July 26, 2018 (the “Prospectus Supplement” and together with the Base Prospectus, the “Prospectus”), relating to the issuance and sale by the Company pursuant to the Registration Statement of 13,244,271 shares of Common Stock (the “Shares”).

The Shares are to be issued and sold by the Company pursuant to an underwriting agreement, dated July 26, 2018 (the “Underwriting Agreement”), by and between the Company and Morgan Stanley & Co. LLC (the “Underwriter”). The Underwriting Agreement has been filed with the Commission as Exhibit 1.1 to the Company’s Current Report on Form 8-K, dated July 31, 2018.

We have acted as counsel for the Company in connection with the issue and sale by the Company of the Shares pursuant to the Underwriting Agreement. We have examined and relied upon corporate or other proceedings of the Company regarding (i) the authorization of (x) the execution and delivery of the Underwriting Agreement, and (y) the issuance of the Shares, (ii) the Registration Statement, (iii) the Base Prospectus, (iv) the Prospectus Supplement, and (v) the Underwriting Agreement. We have also examined and relied upon originals or copies, certified or otherwise identified to our satisfaction, of such other corporate records of the Company, such other agreements and instruments, certificates of public officials and such other documents, instruments and certificates as we have deemed necessary as a basis for the opinions hereinafter expressed.

State Street Corporation

July 31, 2018

Page 2

In our examination of the documents referred to above, we have assumed the genuineness of all signatures, the legal capacity of all individual signatories, the authenticity of all documents submitted to us as originals, the conformity to original documents of all documents submitted to us as copies, the authenticity of such original documents, and the completeness and accuracy of the corporate records of the Company provided to us by the Company.

We express no opinion herein as to the laws of any jurisdiction other than the state laws of the Commonwealth of Massachusetts, the laws of the State of New York and the federal laws of the United States of America.

We express no opinion herein with respect to compliance by the Company with the securities or “blue sky” laws of any state or other jurisdiction of the United States or of any foreign jurisdiction. In addition, we express no opinion and make no statement herein with respect to the antifraud laws or federal antitrust or unfair competition laws of any jurisdiction.

On the basis of, and subject to, the foregoing, we are of the opinion that, when issued and delivered against payment therefor as described in the Underwriting Agreement, the Shares will be validly issued, fully paid and non-assessable.

Please note that we are opining only as to the matters expressly set forth herein, and no opinion should be inferred as to any other matters. This opinion is based upon currently existing statutes, rules, regulations and judicial decisions and is rendered as of the date hereof, and we disclaim any obligation to advise you of any change in any of the foregoing sources of law or subsequent developments in law or changes in facts or circumstances that might affect any matters or opinions set forth herein.

We hereby consent to the filing of this opinion with the Commission as an exhibit to the Company’s Current Report on Form 8-K to be filed on or about July 31, 2018, which Form 8-K will be incorporated by reference into the Registration Statement and to the use of our name therein and in the related Base Prospectus and Prospectus Supplement under the caption “Legal Matters.” In giving such consent, we do not hereby admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act of 1933, as amended, or the rules and regulations of the Commission.

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State Street Corporation

July 31, 2018

Page 3

Very truly yours,