A Londoner, Burrows was born in St Pancras and grew up in Stoke Newington, her mother a trade unionist and primary school teacher in Hackney and her father an architect and teacher. Her parents and stepfather were Socialist Workers Party members, and Burrows was politically active from a young age.[2][3]
Burrows attended William Tyndale Primary School in Islington and then Stoke Newington School. When asked for her preference of a school in Hackney or Hampstead, she said she wanted to be in a multicultural, inclusive environment. She enrolled in acting classes at the Anna Scher Theatre when she was 11.[4][5]
Burrows had a successful modelling career after she was discovered at age 15 in Covent Garden by fashion photographer Beth Boldt.[6][7] For five years she divided her time between London and Paris, where she learned French.[8] She found the modelling world's "obsession with the ideal of the body" troubling.[6][3]
Mike Figgis' ensemble feature Hotel followed, re-uniting Burrows with some colleagues from Timecode, including Salma Hayek and Danny Huston in Venice, where she played the Duchess of Malfi. In 2002, she had a cameo role in Hayek's produced biopic Frida.
She then dedicated herself to stage work. She was directed by Deborah Warner at the Royal National Theatre in Jeanette Winterson's The Powerbook. The play toured, visiting the Theatre National Du Chaillot, Paris, and the Teatro Argentina, Rome. Burrows performed in Spanish in The Galindez File, a film written by Spanish novelist Vazquez Montalban about a woman seeking the truth about the disappearance of a critic of the Dominican dictator Trujillo.[17]
In 2004, she played the part of Andromache in Troy. In January 2005, she created the role of Janey in the world premiere of Earthly Paradise at the Almeida Theatre.[18] The play of a love triangle between Janey Morris; her husband William Morris, the writer and proponent of the Arts and Crafts movement; and Dante Gabriel Rossetti, the Pre-Raphaelite painter.[19] Theatre critic Nicholas De Jongh said of her performance in The Earthly Paradise that "Burrows takes to the stage like a swan to water ... She deserves no end of watching."[20]
On 30 October 2005, she appeared on stage at the Old Vic theatre in London in a rehearsed reading of the 24-hour play Night Sky, alongside Christopher Eccleston.[21][22] In 2006, Burrows was the female lead in the New Zealand thriller Perfect Creature. That same year, she worked with Chilean director Raoul Ruiz on Klimt, his cinematic version of the life of Gustav Klimt. In this film, she played opposite John Malkovich as the artist's lover, a woman of many personalities and nationalities.[11]
In 2008, Burrows starred in the independent film The Guitar, Amy Redford's directorial debut, which premiered at the Sundance Film Festival.[24] In 2008, she had a starring role as Martine Love in Roger Donaldson's heist film The Bank Job.[25] She played opposite Kevin Spacey in Jonas Pate's Shrink. She has contributed to the Actors Come Clean for Congo video for the Raise Hope for Congo campaign, a campaign of the Enough Project, in support of the conflict mineral issue.[26]
She starred in the Amazon Video show Mozart in the Jungle as Cynthia Taylor, a cellist with the New York Symphony. The series ran for four seasons from 2014 to 2018.[33]
Burrows is a Fellow of the Royal Society of Arts.[11] Burrows is bisexual, and has said that she "prefers the company of women". She was engaged to actor Alan Cumming in the 1990s, and dated director Mike Figgis for five years until 2002.[36] She was also previously in a relationship with actress Fiona Shaw.
Burrows married writer Alison Balian, her girlfriend of six years, in August 2013. Burrows gave birth to their son in 2012 and their daughter in 2017.[37][38] They separated in 2020.[39]
Burrows has expressed sympathy for European style social democracies and for French Socialist politicianSégolène Royal.[40] She joined an anti-racism group when she was 11 years old and she went on to become the Vice President of the National Civil Rights Movement.[41] Burrows is a campaigner for disabled rights and equality.[42] In 2009, Burrows became an American citizen.[43]