Burwell was born in New York City, the son of Natalie (née Benedict), a math teacher, and Charles Burwell, who founded Thaibok Fabrics, Ltd. He graduated from King School in Stamford, Connecticut, and Harvard College, where he was a cartoonist for The Harvard Lampoon.
As a film composer, Burwell has had a long-working relationship with the Coen brothers, providing music for every film they have made (except for O Brother, Where Art Thou?, where he provided additional music to a score of traditional songs produced by T-Bone Burnett). He enjoys working with left-field directors, such as Spike Jonze. Among his best known film scores are Miller's Crossing (1990), And the Band Played On (1993), Conspiracy Theory (1997), Hamlet (2000), The Spanish Prisoner (1997), Before the Devil Knows You're Dead (2007), In Bruges (2008), Twilight (2008) Where The Wild Things Are (2009), The Blind Side (2009), and Breaking Dawn Part 1 (2011).
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