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.

Mays starred in two Broadway revivals in 2007. He appeared as Henry Higgins in Pygmalion, in which he won critical praise. The TheaterMania reviewer called him "pitch-perfect", and the production an "excellent revival." He appeared in a revival of Journey's End in February to June 2007, in which he starred as Private Mason. Ben Brantley, in the New York Times, called the production "acutely staged and acted." The TheaterMania reviewer wrote: "Mays is a reliable, obliging Mason."

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 performed in the musical from October 2013 (previews) until closing on January 17, 2016. 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).