Exhibit 5.1

Curis, Inc.

4 Maguire Road

Lexington, MA 02421

Ladies and Gentlemen:

This opinion is furnished to you in connection with (i) the Registration Statement on Form S-3 (File No. 333-224627) (the “Registration Statement”) filed by Curis, Inc., a Delaware corporation (the “Company”), with the Securities and Exchange Commission (the “Commission”) under the Securities Act of 1933, as amended (the “Securities Act”), for the purpose of registering with the Commission under the Securities Act, among other things, an indeterminate number of shares of common stock, $0.01 par value per share, of the Company (the “Common Stock”), all of which may be issued and sold by the Company from time to time on a delayed or continuous basis pursuant to Rule 415 under the Securities Act at an aggregate initial offering price not to exceed $200,000,000, as set forth in the Registration Statement and the prospectus contained therein (the “Base Prospectus”); and (ii) the prospectus supplement, dated March 6, 2020 (the “Prospectus Supplement” and together with the Base Prospectus, the “Prospectus”), relating to the issuance and sale from time to time by the Company of shares of Common Stock with an aggregate offering price of up to $30,000,000 (the “Sales Agreement Shares”).

The Sales Agreement Shares are to be issued and sold by the Company pursuant to a Capital on Demand^TM Sales Agreement, dated March 4, 2020, between the Company and JonesTrading Institutional Services LLC (the “Sales Agreement”). The Sales Agreement is being filed with the Commission as Exhibit 1.1 to the Company’s Current Report on Form 8-K, filed on the date hereof.

We are acting as counsel for the Company in connection with the issuance and sale by the Company of the Sales Agreement Shares. We have examined and relied upon copies of the Registration Statement and the Prospectus as filed with the Commission, including the exhibits thereto. We have also examined and relied upon the Sales Agreement, the Certificate of Incorporation of the Company (as amended or restated from time to time, the “Certificate of Incorporation”), the By-laws of the Company (as amended or restated from time to time, the “By-laws”), the minutes of meetings and actions of the stockholders and the Board of Directors of the Company, including committees thereof as provided to us by the Company, and such other documents as we have deemed necessary for purposes of rendering the opinions hereinafter set forth.

In our examination of the foregoing documents, we have assumed the genuineness of all signatures, 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 the originals of

Wilmer Cutler Pickering Hale and Dorr LLP, 60 State Street, Boston, Massachusetts 02109

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Curis, Inc. March 6, 2020 Page 2

such latter documents and the legal capacity of all signatories to such documents. Further, we have assumed that the Company will not issue and sell pursuant to the Sales Agreement such number of Shares that would cause the Company not to satisfy the eligibility requirements for Form S-3 (including, as applicable, Instruction I.B.6. thereof).

We express no opinion herein as to the laws of any state or jurisdiction other than the state laws of the Commonwealth of Massachusetts, the General Corporation Law of the State of Delaware and the federal laws of the United States of America.

Based upon and subject to the foregoing, we are of the opinion that the Sales Agreement Shares have been duly authorized for issuance and, when the Sales Agreement Shares are issued and paid for in accordance with the terms and conditions of the Sales Agreement, the Shares will be validly issued, fully paid and nonassessable.

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 we disclaim any obligation to advise you of any change in any of these sources of law or subsequent legal or factual developments which 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 Current Report on Form 8-K to be filed by the Company on the date hereof in connection with the issuance and sale of the Shares in accordance with the requirements of Item 601(b)(5) of Regulation S-K under the Securities Act and to the use of our name therein and in the related 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 or the rules and regulations of the Commission.

Very truly yours,

WILMER CUTLER PICKERING

HALE AND DORR LLP