Mays was raised in Clinton, Connecticut with his parents, a naval intelligence officer and a children's librarian, and his siblings. Mays graduated from Yale College, where he received a Bachelor of Arts degree, and the University of California, San Diego, where he earned a Master of Fine Arts.
Mays appeared on Broadway in I Am My Own Wife, a Pulitzer Prize-winning play by Doug Wright, from November 2003 (previews) to October 31, 2004. He had appeared in the play Off-Broadway at Playwrights Horizons in May 2003, and at the La Jolla Playhouse in July 2001.
Mays won the 2004 Tony Award for Best Performance by a Leading Actor in a Play, the 2004 Drama Desk Award for Outstanding One-Person Show, an Obie Award, and a 2004 Theatre World Award for his solo performance. He also won the 2007 Helpmann Award for Best Male Actor in a Play for seasons of I Am My Own Wife in Australia in 2006.
In 2007, he appeared as Henry Higgins in a revival of Pygmalion and starred as Private Mason in a revival of Journey's End. In August 2009, Mays appeared at the Williamstown Theatre Festival in Quartermaine's Terms by Simon Grey.
Mays starred in the 2013 Broadway musical A Gentleman's Guide to Love and Murder, in which he played eight roles. He won the Outer Critics Circle Award for Outstanding Actor in a Musical. He also was nominated for the Tony Award for Best Performance by a Leading Actor in a Musical and tied for the Drama Desk Award for Outstanding Actor in a Musical (with Neil Patrick Harris).