Exhibit 99.1

Kura Oncology Announces Commencement of Public Offering of Common Stock

SAN DIEGO, Calif., June 27, 2018 – Kura Oncology, Inc. (Nasdaq: KURA), a clinical-stage biopharmaceutical company focused on the development of precision medicines for oncology, today announced that it has commenced an underwritten public offering, subject to market and other conditions, to issue and sell shares of its common stock. In connection with the offering, Kura Oncology expects to grant the underwriters a 30-day option to purchase up to an additional 15% of the shares of its common stock offered in the public offering. There can be no assurance as to whether or when the offering may be completed, or as to the actual size or terms of the offering.

Leerink Partners is acting as sole book-running manager in the offering.

The securities described above are being offered by Kura Oncology pursuant to a shelf registration statement on Form S-3, including a base prospectus, that was previously filed by Kura Oncology with the Securities and Exchange Commission (the “SEC”) and that was declared effective on December 4, 2017. A preliminary prospectus supplement and accompanying prospectus relating to the offering will be filed with the SEC and will be available for free on the SEC’s website located at http://www.sec.gov. Copies of the preliminary prospectus supplement and the accompanying prospectus relating to the offering, when available, may be obtained from Leerink Partners LLC, Attention: Syndicate Department, One Federal Street, 37th Floor, Boston, MA 02110, or by telephone at (800) 808-7525, ext. 6132, or by email at syndicate@leerink.com.

This press release shall not constitute an offer to sell or the solicitation of an offer to buy these securities, nor shall there be any sale of these securities in any state or other jurisdiction in which such offer, solicitation or sale would be unlawful prior to the registration or qualification under the securities laws of any such state or other jurisdiction.

About Kura Oncology

Kura Oncology is a clinical-stage biopharmaceutical company committed to realizing the promise of precision medicines for the treatment of cancer. Kura Oncology’s pipeline consists of small molecule drug candidates that target cancer signaling pathways where there is a strong scientific and clinical rationale to improve outcomes by identifying those patients most likely to benefit from treatment. Kura Oncology’s lead drug candidate is tipifarnib, a farnesyl transferase inhibitor, which is currently being studied in multiple Phase 2 clinical trials in solid tumor and hematologic indications. Kura Oncology’s pipeline also includes KO-947, an ERK inhibitor, currently in a Phase 1 dose-escalation trial, and KO-539, a menin-MLL inhibitor, currently in preclinical development.

Forward-Looking Statements

Statements contained in this press release regarding matters that are not historical facts are “forward-looking statements” within the meaning of the Private Securities Litigation Reform Act of 1995. Such forward-looking statements include, without limitation, statements about Kura Oncology’s expectations regarding the completion, timing and size of the proposed offering, and

its expectations with respect to granting the underwriters a 30-day option to purchase additional shares. Because such statements are subject to risks and uncertainties, actual results may differ materially from those expressed or implied by such forward-looking statements. Words such as “believes,” “anticipates,” “plans,” “expects,” “intends,” “will,” “goal,” “potential” and similar expressions are intended to identify forward-looking statements. These forward-looking statements are based upon Kura Oncology’s current expectations and involve assumptions that may never materialize or may prove to be incorrect. Actual results and the timing of events could differ materially from those anticipated in such forward-looking statements as a result of various risks and uncertainties, which include, without limitation, risks associated with market conditions and the satisfaction of closing conditions related to the proposed public offering, as well as risks and uncertainties associated with Kura Oncology’s business and finances in general, and the other risks described in Kura Oncology’s quarterly report on Form 10-Q for the quarterly period ended March 31, 2018 and other filings with the SEC. All forward-looking statements contained in this press release speak only as of the date on which they were made. Kura Oncology undertakes no obligation to update such statements to reflect events that occur or circumstances that exist after the date on which they were made.

Contacts

Company:

Pete De Spain

Vice President, Investor Relations &

Corporate Communications

(858) 500-8803

pete@kuraoncology.com

Investors:

Robert H. Uhl

Managing Director

Westwicke Partners, LLC

(858) 356-5932

robert.uhl@westwicke.com

Media:

Jason Spark

Managing Director

Canale Communications

(619) 849-6005

jason@canalecomm.com