He had piano lessons as a boy. Although he was talented, the lessons themselves did not spark a strong interest in making music. That occurred in high school, when a friend taught him how to improvise blues on piano. Burwell discovered a joy in creating music that led him to continuing blues and jazz playing and to compose.
He went to Harvard as a Fine Arts major. While there, he continued to play music wherever he could. In his final year at the University, he joined a band, more or less as a recreational activity.
One evening, the sound editor on a small independent film written and directed by an unknown fledgling pair of brothers happened to hear Burwell in his band. The brothers were Ethan and Joel Coen, and the film they were making was called Blood Simple. It was a dark, moody murder story with sudden, even brutal drops into blood and violence. Burwell was asked to score it and responded with an entirely non-Hollywood-sounding score for just four musicians: himself on keyboards plus bass guitar and two percussionists.
The film was a critical success and was profitable. The Coen Brothers' next film was very different in mood, and became a cult favorite. For this film, Raising Arizona (1987), Burwell sketched the musical counterparts of its odd group of characters with another small group, again including himself on synthesizers and samplers, banjo, three ukeleles, two percussionists, a whistling guitarist, and a yodeler. The film's success was astonishing, but no less so than the remarkable appropriateness of the spirited music.
Burwell has continued to be the composer of choice for the Coen Brothers. He has scored Miller's Crossing (1990), Barton Fink (1991), The Hudsucker Proxy (1994), Fargo (1996), and The Big Lebowski (1998). He came into demand as a film score composer, and has written for diverse genres including horror (Psycho III, Buffy the Vampire Slayer), drama (The Spanish Prisoner, This Boy's Life, Gods and Monsters, The Chamber), comedy (Wayne's World 2, Airheads, Joe's Apartment), romantic adventure (Rob Roy), political thriller (The Jackal, Fear, Conspiracy Theory), suspense (Scorchers, Fear), and romantic comedy (Doc Hollywood, Picture Perfect, It Could Happen to You). In addition to his work with the Coen Brothers, Burwell also has worked three times with director Michael Caton-Jones (Roy Boy, This Boy's Life, and Doc Hollywood).
In fifteen years, he wrote music for more than forty-five films, with demand for his work rising: He composed for seven films in 1995, six in 1996, seven in 1997, and five in 1998. His scores have a knack for evoking intimate feelings, especially loneliness and desperation.
He also composed for the television project And the Band Played On and has written music for the New York stage productions Mother, The Celestial Alphabet Event, and Mother Courage, and music for Widows at the Williamstown Theatre Festival.