Kura Oncology (KURA) 8-K Kura Oncology Appoints Carol Schafer to

Exhibit 99.1

Kura Oncology Appoints Carol Schafer to

Board of Directors

SAN DIEGO, June 4, 2021 – Kura Oncology, Inc. (Nasdaq: KURA), a clinical-stage biopharmaceutical company committed to realizing the promise of precision medicines for the treatment of cancer, today announced the appointment of Wall Street veteran Carol Schafer to its board of directors, effective June 3, 2021. Ms. Schafer has more than 25 years of experience in healthcare investment banking, most recently as Vice Chair of Equity Capital Markets at Wells Fargo Securities.

“Carol brings a wealth of experience as a trusted strategic and financial advisor to the leadership teams of growing biopharmaceutical companies,” said Troy Wilson, Ph.D., J.D., President and Chief Executive Officer of Kura Oncology. “We welcome her to the board and look forward to her contributions as we work to bring our oncology drug candidates to market, expand their use to larger patient populations and create value for patients and shareholders.”

Ms. Schafer currently serves as a Managing Partner at Hyphen Advisors. She previously served as Vice Chair of Equity Capital Markets at Wells Fargo Securities, where she oversaw more than 200 transactions that raised nearly $90 billion in proceeds. Prior to that, Ms. Schafer served as Vice President of Finance and Business Development at Lexicon Pharmaceuticals. Earlier, she held positions of increasing responsibility at J.P. Morgan, most recently as Managing Director of Equity Capital Markets. Ms. Schafer currently serves on the board of directors of Insmed, Idera Pharmaceuticals and Repare Therapeutics, and she previously served on the board of Five Prime Therapeutics until its acquisition by Amgen in April 2021. She received a B.A. from Boston College and an M.B.A from New York University.

“Kura has a promising pipeline of oncology drug candidates, an impressive team and the financial resources to create significant value,” said Ms. Schafer. “I look forward to working with the leadership team and my fellow board members to help the company achieve its strategic objectives and realize this value.”

Ms. Schafer will succeed Robert Hoffman, who will be resigning from the board effective August 3, 2021. Ms. Schafer will also succeed Mr. Hoffman as Audit Committee Chair at that time. Mr. Hoffman has served on Kura’s board since March 2015.

“I want to take this opportunity to express my sincere appreciation for Robert’s service and his many contributions throughout his six years on the Kura board,” continued Dr. Wilson. “We are deeply grateful for his service as a board member and audit committee chair, and on behalf of everyone at Kura, we wish him well in his future endeavors.”

About Kura Oncology

Kura Oncology is a clinical-stage biopharmaceutical company committed to realizing the promise of precision medicines for the treatment of cancer. The Company’s pipeline consists of small molecule drug candidates that target cancer signaling pathways. KO-539, a potent and selective menin inhibitor, is currently in a Phase 1/2 clinical trial (KOMET-001) and targeting patients with relapsed/refractory acute myeloid leukemia, including patients with NPM1 mutations. Tipifarnib, a potent, selective and orally bioavailable farnesyl transferase inhibitor, has received Breakthrough Therapy Designation for the treatment of patients with HRAS mutant head and neck squamous cell carcinoma and is currently in a registration-directed study (AIM-HN) in patients with this devastating disease. Kura is also developing a next-generation farnesyl transferase inhibitor, which is intended to target innovative biology and larger oncology indications through rational combinations. For additional information about Kura, please visit the Company’s website at www.kuraoncology.com.

Forward-Looking Statements

This news release contains certain forward-looking statements that involve risks and uncertainties that could cause actual results to be materially different from historical results or from any future results expressed or implied by such forward-looking statements. Such forward-looking statements include statements regarding, among other things, the efficacy, safety and therapeutic potential of Kura’s drug candidates, tipifarnib and KO-539, progress and expected timing of Kura’s drug development programs and clinical trials and submission of regulatory filings, the presentation of data from clinical trials, plans regarding regulatory filings and future clinical trials, the regulatory approval path for tipifarnib, the strength of Kura’s balance sheet and the adequacy of cash on hand. Factors that may cause actual results to differ materially include the risk that compounds that appeared promising in early research or clinical trials do not demonstrate safety and/or efficacy in later preclinical studies or clinical trials, the risk that Kura may not obtain approval to market its drug candidates, uncertainties associated with performing clinical trials, regulatory filings, applications and other interactions with regulatory bodies, the risks associated with reliance on third parties to successfully conduct clinical trials, the risks associated with reliance on outside financing to meet capital requirements, the risks associated with the COVID-19 global pandemic, and other risks associated with the process of discovering, developing and commercializing drugs that are safe and effective for use as human therapeutics, and in the endeavor of building a business around such drugs. You are urged to consider statements that include the words “may,” “will,” “would,” “could,” “should,” “believes,” “estimates,” “projects,” “promise,” “potential,” “expects,” “plans,” “anticipates,” “intends,” “continues,” “designed,” “goal,” or the negative of those words or other comparable words to be uncertain and forward-looking. For a further list and description of the risks and uncertainties the Company faces, please refer to the

Company’s periodic and other filings with the Securities and Exchange Commission, which are available at www.sec.gov. Such forward-looking statements are current only as of the date they are made, and Kura assumes no obligation to update any forward-looking statements, whether as a result of new information, future events or otherwise.

Contacts

Company:

Pete De Spain

Vice President, Investor Relations &

Corporate Communications

(858) 500-8803

pete@kuraoncology.com

Investors:

Robert H. Uhl

Managing Director

Westwicke ICR

(858) 356-5932

robert.uhl@westwicke.com

Media:

Jason Spark

Managing Director

Canale Communications

(619) 849-6005

jason@canalecomm.com