Crowley trains at the Bristol Old Vic Theatre School in Bristol, England. He designs over twenty productions for the Royal National Theatre in London including Ghetto, The Madness of George III, Carousel and The History Boys. He also designs numerous productions for the Royal Shakespeare Company including The Plantagenets, for which he wins a Laurence Olivier Award for Best Set Design, and Les Liaisons Dangereuses, which later has a successful run in London, followed by a transfer to Broadway. Opera productions include the critically acclaimed production of The Magic Flute directed by Nicholas Hytner, with whom Crowley frequently collaborates, for the English National Opera and La traviata for the Royal Opera House.
Crowley has received multiple Tony Award nominations, and has won seven times, for designing the Broadway productions of Carousel (1994), Aida (2000), The History Boys (2006), Mary Poppins (2007), The Coast of Utopia (2007), Once (2012) and An American in Paris (2015). He receives three other Tony Award nominations in 2015, two for his costumes on The Audience and An American in Paris and one for his scenic designs for Skylight. He is a recipient of the Laurence Olivier Award for Best Set Design and a three-time recipient of the Drama Desk Award for Outstanding Set Design.
Crowley designs set and costume for Mary Poppins, which plays in both the West End theatre and on Broadway. He designs and directs the Phil Collins musical Tarzan. He is the set and costume designer for Andrew Lloyd Webber‘s Love Never Dies, and the costume designer of the 2008-2009 version of The Little Mermaid.