Curran was born in Glasgow, Scotland. He is an alumnus of Holyrood Secondary School and is a graduate of the Royal Scottish Academy of Music and Drama.
Tony rose to fame in the BBC television series This Life-playing a gay plumber. Since then, the Scots star has appeared in a number of major film and television roles.
To play The Invisible Man in The League of Extraordinary Gentlemen, he donned a special suit that turned him into a walking bluescreen (according to his commentary on the DVD, he looked like a "smurf on acid"). He has played murderers, vampires, Vikings, leprechauns, and 'regular guys'.
Curran is active in participating in marathons to raise money for charity and is currently in a relationship. He is a keen fan of Celtic Football Club.
He is a frequent and popular participant in the annual Dressed to Kilt event in New York City, run by the organization Friends of Scotland in celebration of Tartan Week.
In April, 2009 Curran revealed to the Daily Record newspaper that he is playing the role of Lieutenant Delcourt, in The Adventures of Tintin: Secret of the Unicorn, due to be released in 2011. Curran also played Dutch painter Vincent Van Gogh, in the Doctor Who episode "Vincent and the Doctor", written by Richard Curtis.