Greig was born in Maidstone, Kent, the second of three sisters. Her father, Eric, worked as a colour Chemist creating dyes, and her mother, Ann, was enthusiastic about amateur dramatics. Her Grandfather was a Polish Jew. The family moved to Kilburn when she was three. She went to Malorees Junior School, followed by Camden School for Girls where she passed A-Levels in English, French and Mathematics, and graduated with a first in Drama and Theatre Arts from the University of Birmingham (BA in Drama and Theatre Arts 1988). After graduating, she worked at the Family Planning Association and continued doing temporary work until 1996. She also spent some time at a secretarial college.

Her career began in early 1990, Greig is known for both dramatic and comedic roles. Comedy roles generally pose problems for Greig, who has admitted that she has problems with corpsing.

Greig has had a long-running part as Debbie Aldridge in the BBC Radio 4 soap opera The Archers since 1991. As her other work increased, her appearances in the show decreased and her character Debbie spends most of her time living in Hungary.

Greig lives in a flat in Kensal Green, having moved back in 1996 to be with her dying father. She became a Christian at this time, having been brought up as an atheist. Greig is also a vegetarian. She has been married to actor Richard Leaf since 1997, whom she met on the set of Neil Gaiman's 1996 miniseries Neverwhere, and has three children.

Greig starred with Richard E. Grant in the film Cuckoo, and with Roger Allam and Gemma Arterton in Tamara Drewe. She also made a short cameo appearance in the 2004 comedy Shaun of the Dead, and co-starred in 2015's The Second Best Exotic Marigold Hotel. 2016 saw the release of Breaking The Bank, which sees her portray Penelope Bunbury opposite Kelsey Grammer as Charles Bunbury.

She starred in the BBC comedy drama series Love Soup (2005), as Alice Chenery, a lovelorn woman working on a department store perfume counter, in a role specifically written for her by David Renwick, whom she met in 2003 when she appeared in an episode of Jonathan Creek. In May 2005 she also appeared as a nurse in an episode of the BBC series Doctor Who, entitled "The Long Game".

During 2006 and early 2007, Greig played Beatrice in a much acclaimed production of Much Ado About Nothing for which she won a Laurence Olivier Award, and Constance in King John, as part of the Royal Shakespeare Company's The Complete Works season. Whilst the win itself was a surprise, her acceptance speech was received very well as being highly entertaining, claiming that she was so excited that she had wet her dress. The speech was apparently completely improvised. Backstage, when told not to tell her mother about her wetting her dress, she told the host that her mum was dead before dedicating her award to her "dead mum". She also won the Critics' Circle Theatre Award for "Best Shakespearean Performance" in Much Ado About Nothing, becoming the first woman to win the award, and was nominated for "The FRANCO'S Best Actress in a Play" in the Whatsonstage Theatregoers' Choice Awards.

At the Gielgud Theatre in March 2008, she co-starred with Ralph Fiennes, Janet McTeer and Ken Stott in the UK premiere of Yasmina Reza's The God of Carnage (Le Dieu du Carnage) translated by Christopher Hampton and directed by Matthew Warchus. The play won the Laurence Olivier Award for Best New Comedy in 2009. In 2008, she co-starred in the surreal sci-fi film Captain Eager and the Mark of Voth.

Greig appeared in the role of Edith Frank in the BBC's January 2009 production of The Diary of Anne Frank. Also in 2009, she appeared as Miss Bates in the BBC serial Jane Austen's Emma. In 2010, she played Sacharissa Cripslock in the two part mini-series Terry Pratchett's Going Postal.

In November 2008, she made her National Theatre debut in Gethsemane, a new play by David Hare which toured the UK. Greig was starring in The Little Dog Laughed by Douglas Carter Beane at the Garrick Theatre in London, which did run a limited season until 10 April 2010. She starred alongside Rupert Friend, Gemma Arterton and Harry Lloyd, and the play was directed by Jamie Lloyd. She won the 2011 WhatsOnStage Theatregoers Choice Award for Best Supporting Actress in a play for her portrayal. Her performance as Diane in The Little Dog Laughed garnered her a second Olivier Award nomination for Best Actress. In October 2011 she was Hilary, the central character, in Jumpy at the Royal Court, London., which later transferred to the Duke of York's Theatre in the West End. In March 2013 she played Varia in Longing, a new play by william Boyd based on two short stories by Chekov, at the Hampstead Theatre. Greig previously performed in Women on the Verge of a Nervous Breakdown at the Playhouse Theatre, London, until May 2015. In March 2015, she received a nomination for the Laurence Olivier Award for Best Actress in a Musical.

In 2011, she starred in the BBC/Showtime sitcom Episodes, alongside Matt LeBlanc and Green Wing co-star Stephen Mangan. Greig and Mangan play a husband-and-wife writing duo who travel to America to work on an adaptation of their successful series. Greig also stars in the Channel 4 sitcom, Friday Night Dinner, as the mother of a North London Jewish family.

She played Beth in the 2012 BBC series White Heat. She also is the lead in The Guilty in the 3 part series on ITV in 2013, playing DCI Maggie Brand investigating the death of a young child who went missing five years ago. In 2014, she played Sally in the Inside No. 9 episode "Last Gasp".

She is a supporter of the National Health Service, giving her backing to a rally organised by pro-NHS protest group NHS Together. She also supports more practical teaching of Shakespeare in British schools, supporting the RSC's "Stand Up For Shakespeare" manifesto. In August 2014, Greig was one of 200 public figures who were signatories to a letter to The Guardian opposing Scottish independence in the run-up to September's referendum on that issue.

In 2015 the fourth season of Episodes was aired, and in 2016 a seven-episode fifth series, still starring Greig alongside Stephen Mangan and Matt LeBlanc, was filmed in London.

In October 2016, she returned to the Hampstead Theatre to play Empty in The Intelligent Homosexual's Guide to Capitalism and Socialism with a Key to the Scriptures by Tony Kushner. In February 2017 she returned to the Royal National Theatre to play Malvolia in a new production of Twelfth Night at the Olivier Theatre. As a Labour constituency agent spanning a period of twenty-seven years, she gave a "polished…magnificent" performance in James Graham's Labour of Love at the Noel Coward Theatre, London, in October 2017.