Exhibit 5.1

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 “Registrant”), 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, shares of common stock, $0.01 par value per share (the “Common Stock”), of the Registrant, all of which may be issued and sold by the Registrant 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; and (ii) the prospectus supplement, dated February 26, 2020 (the “Prospectus Supplement”), relating to the issuance and sale by the Registrant of shares of Common Stock with an aggregate offering price of up to $30,000,000 (the “Purchase Shares”) and an additional 646,551 shares of Common Stock (the “Commitment Shares” and, together with the Purchase Shares, the “Shares”).

The Purchase Shares are to be issued and sold by the Registrant pursuant to a Common Stock Purchase Agreement, dated February 26, 2020 (the “Purchase Agreement”), between the Registrant and Aspire Capital Fund, LLC (“Aspire”). The Commitment Shares are to be issued by the Registrant to Aspire pursuant to the Purchase Agreement in consideration for entry into the Purchase Agreement.

We are acting as counsel for the Registrant in connection with the issuance and sale by the Registrant of the Shares. We have examined a signed copy of the Registration Statement as filed with the Commission, including the exhibits thereto, and the Prospectus Supplement as filed with the Commission. We have also examined and relied upon the Purchase Agreement, the minutes of meetings of the stockholders and the Board of Directors of the Registrant as provided to us by the Registrant, the Certificate of Incorporation and Bylaws of the Registrant, each as restated and/or amended to date, and such other documents as we have deemed necessary for purposes of rendering the opinion 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 such latter documents and the legal competence of all signatories to such documents.

We express no opinion herein as to the laws of any state or jurisdiction other than the General Corporation Law of the State of Delaware.

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

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

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Curis, Inc. February 26, 2020 Page 2

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 Registrant 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 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.