Acting
Burrows made her film debut in 1993 with a small role in Jim Sheridan's In the Name of the Father. Her first significant acting roles came in 1995, as an ambitious young Irishwoman in Circle of Friends, and in Ngozi Onwurah's Welcome II The Terrordome. In 1996 she starred in the BBC production of Dennis Potter's Karaoke and in Hotel de Love; the next year she was seen in Lovelife, Nevada, The Matchmaker and Mike Figgis's One Night Stand. In 1999 she appeared in Figgis's experimental film The Loss of Sexual Innocence, in which she played twins – one raised in England, the other in Italy. In 1999, she appeared in the thriller Deep Blue Sea, and had the title role in Figgis's film adaptation Miss Julie, which premiered at the Toronto International Film Festival. She starred for Figgis again alongside Stellan Skarsgård in Timecode (2000), a split-screen digital experimental film shot in a single take with no edits. She followed it with Gangster No. 1, starring Malcolm McDowell, Paul Bettany and David Thewlis. She next co-starred with Kate Winslet and Dougray Scott in Michael Apted's 1940s drama Enigma and starred in Tempted, an improvised thriller set in New Orleans, with Burt Reynolds and Peter Facinelli.
Figgis's ensemble feature Hotel followed, reuniting Burrows with colleagues from Timecode including Salma Hayek and Danny Huston on location in Venice, where she played the Duchess of Malfi. She had a cameo in Frida, Hayek's 2002 biopic of artist Frida Kahlo.
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. In 2004, she played the role of Andromache, the wife of Trojan Prince Hector, played by Eric Bana, in Troy.
Burrows dedicated herself to stage work in the early 2000s. She appeared at the Royal National Theatre in Jeanette Winterson's The Powerbook, directed by Deborah Warner; the play also went on tour, visiting the Theatre National Du Chaillot in Paris and the Teatro Argentina in Rome.
In January 2005, she created the role of Janey Morris in the world premiere of Earthly Paradise at the Almeida Theatre. The play centered on the love triangle of Janey, her husband William Morris (the writer and proponent of the Arts and Crafts movement), and the Pre-Raphaelite painter Dante Gabriel Rossetti. London theatre critic Nicholas De Jongh said of her performance: "Burrows takes to the stage like a swan to water ... She deserves no end of watching." 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. Burrow has since participated in the "24 Hour Plays" in New York and Los Angeles.
In 2006, Burrows was the female lead in the New Zealand thriller Perfect Creature, appeared in Hal Hartley's film Fay Grim, and co-starred in Klimt, Chilean director Raoul Ruiz' cinematic portrait of Gustav Klimt. Burrows played opposite John Malkovich as the artist's lover, a woman of many personalities and nationalities. Onstage in 2006, she starred opposite David Schwimmer in the world premiere of Neil LaBute's Some Girl(s) at the Gielgud Theatre, London. She then appeared in Mike Binder's film Reign Over Me with Don Cheadle and Adam Sandler. Burrows also had lead roles in the Indian thriller Broken Thread and in director Peter Howitt's drama Dangerous Parking. On television, she played attorney Lorraine Weller on ABC's Boston Legal (season 4) from 2007 to 2008. She starred in the NBC series My Own Worst Enemy in 2008.
Also in 2008, Burrows starred in the independent film The Guitar as a woman who suffers several life setbacks all at once, including a cancer diagnosis and bankruptcy, and survives them all. Amy Redford's directorial debut, the film premiered at the Sundance Film Festival. She also had a starring role in Roger Donaldson's heist film The Bank Job, and appeared opposite Kevin Spacey in Jonas Pate's Shrink. She contributed to an Actors Come Clean for Congo video for the Enough Project's campaign in support of conflict mineral issues. In 2010, she starred as Detective Serena Stevens on Law & Order: Criminal Intent, departing at the end of the ninth season. In September 2010, she took part in the documentary feature The People Speak, televised on the History Channel. The film was directed and produced by Colin Firth and Anthony Arnove. Burrows modelled for Marks & Spencer's autumn 2010 campaign for their Portfolio range. In 2012, she performed opposite Rob Lowe in the political comedy Knife Fight.
In 2013 and 2014, Burrows joined the ABC television show Agents of S.H.I.E.L.D., part of the Marvel Cinematic Universe. She played agent Victoria Hand in a recurring role. She also starred in the Amazon Video series Mozart in the Jungle as Cynthia Taylor, a cellist with the New York Symphony. The series ran for four seasons from 2014 to 2018. In 2019, Burrows starred in the recurring role of Dottie Quinn on the second season of the Netflix thriller You. She reprised her role in the third season, which was released in October 2021.
