Bremner was born in Edinburgh, a son of two art teachers. His father's name is Aidan. He attended Davidson's Mains Primary School and Portobello High School. He originally wanted to be a circus clown, but was offered a chance in show-business by television director, Richard D. Brooks. One of his first notable roles was as a Glasgow schoolboy in Charles Gormley's Heavenly Pursuits (1985). He would also play the lead in the BBC Scotland feature-length adaptation of the William McIlvanney short story 'Dreaming' (1990).

Bremner is probably best known to British audiences for his portrayal of Spud in Danny Boyle's film adaptation of Irvine Welsh's Trainspotting, and Mullet, the wimpish street thug in Guy Ritchie's Snatch. In Trainspotting, Bremner played the childlike, likeable, gullible and seemingly doomed Spud to excellent reviews, and in Snatch is probably best remembered for his memorable interrogation scene with Vinnie Jones.

In the 1995 stage version of Trainspotting, Bremner played the lead role of Renton, the role played by Ewan McGregor in the film. In America, he is probably best known for playing supporting roles in blockbusters such as Pearl Harbor and Black Hawk Down, in which he appeared alongside fellow Trainspotting star Ewan McGregor. In 2005, he had a small role in the mini-series Elizabeth I as James VI, King of Scots. In December 2006, he played Harold Stritzke in the Sci Fi Channel miniseries The Lost Room.

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