He was educated at Northgate Grammar School, Ipswich and Downing College, Cambridge. At Northgate, he had an inspiring English Teacher, Peter Hewett, who also directed the school plays. Hewett encouraged him to sit the scholarship exam in Cambridge in the hope of studying under F R Leavis at Downing. Hewett also persuaded the headmaster to help with the cost of Nunn staying in Cambridge to take the exam. Nunn's father could not afford it and the headmaster had refused at first so Nunn was close to giving up. At Downing, Nunn began his stage career and first met contemporaries Ian McKellen and Derek Jacobi. In 1962, he directed Macbeth for The Marlowe Society and he directed that year's Footlights. He also won a Director's Scholarship, becoming a trainee Director at the Belgrade Theatre in Coventry.

In 1964, Nunn joined the Royal Shakespeare Company and in 1968 he was appointed Artistic Director of the RSC, a position he held until 1986 (latterly with Terry Hands from 1978).

Nunn has been married three times. He was married to Actress Janet Suzman from 17 October 1969 until their divorce in 1986. They have one child, Joshua. The Times reported in April 1986: "Janet Suzman, aged 47, the Actress, was granted a divorce in London yesterday from Trevor Nunn, aged 46, the theatre Director ... They have a son, Joshua, aged five." He has another two children, Laurie and Amy, with his second wife, Sharon Lee-Hill; they divorced in 1991. According to The Times, "Trevor Nunn, the theatre Director, was divorced yesterday by his wife Sharon Lee Hill."

His first wife, Janet Suzman, appeared in many of his productions, such as the 1974 televised version of his Antony and Cleopatra. Nunn directed the RSC production of Macbeth starring Ian McKellen in the title role and Dame Judi Dench as Lady Macbeth in 1976. Nunn staged the action of the drama with not only the paying audience, but also the audience of all of the actors in the production not in the ongoing scene—they sat on wooden crates just beyond the main playing space.

Nunn became a leading figure in theatrical circles, and was responsible for many significant productions, such as the RSC's version of Dickens's Nicholas Nickleby, co-directed with John Caird, and a 1976 musical adaptation of the Shakespeare play The Comedy of Errors.

A very successful Director of musicals, in the non-subsidised sector, Nunn directed the musical Cats (1981), formerly the longest running musical in Broadway's history, and the first English production of Les Misérables in 1985, also with John Caird, which has been running continuously in London since opening. Nunn also directed the little-known 1986 Webber–Rice musical Cricket, at Windsor Castle. Besides Cats and Les Misérables Nunn's other musical credits include Starlight Express and Sunset Boulevard. He became Artistic Director of the Royal National Theatre in September 1997, a position he retained until 2003.

He has directed for film, including Lady Jane (1986), Hedda, an adaptation of Hedda Gabler, and a 1996 film version of Shakespeare's Twelfth Night.

Nunn has directed opera at Glyndebourne. He re-staged his highly successful Glyndebourne production of Gershwin's Porgy and Bess for television in 1993, and was highly praised.

In 1994, he married Actress Imogen Stubbs with whom he has two children, Ellie and Jesse. The Press Association reported on 18 September 1994: "Director Trevor Nunn and Actress Imogen Stubbs, who wed in secret at a London register office on Saturday, had their marriage blessed today in a low-key Service at a village church." In April 2011 Stubbs announced that she and Sir Trevor were to separate.

In 1998 Nunn was named in a list of the biggest private financial donors to the Labour Party. In 2002, he was knighted.

He directed We Happy Few, a play by his second wife Imogen Stubbs, in 2004. Stubbs often appears in his productions, including the 1996 Twelfth Night film. Nunn directed a modern production of Shakespeare's Hamlet in 2004, which starred Ben Whishaw in the title role, and Imogen Stubbs as Gertrude, and was staged at the Old Vic Theatre in London.

Later London credits include My Fair Lady, South Pacific (at the Royal National Theatre),The Woman in White,Othello and Acorn Antiques: The Musical! (2005),The Royal Hunt of the Sun, Rock 'n' Roll and Porgy and Bess in 2006 at the Savoy Theatre (an abridged version with dialogue instead of recitatives, unlike Nunn's first production of the opera).

In 2007, he directed the RSC productions of King Lear and The Seagull, which played at Stratford before embarking on a world tour (including the Brooklyn Academy of Music) and then playing at the New London Theatre from November 2007. The two plays both starred Ian McKellen, Romola Garai, Frances Barber, Sylvester McCoy, and william Gaunt. Nunn's television production of King Lear was screened on Boxing Day, 2008 with McKellen in the title role.

In 2008, he returned to The Belgrade Theatre in Coventry (the theatre where he started his career) to direct Joanna Murray-Smith's adaptation of Ingmar Bergman's film Scenes from a Marriage starring Imogen Stubbs and Iain Glen. His musical adaptation of Gone with the Wind opened at the New London Theatre in April 2008 and, after poor reviews, closed on 14 June 2008 after 79 performances. In December 2008, he directed a revival of A Little Night Music at the Menier Chocolate Factory, which transferred to the West End at the Garrick Theatre in 2009. The production transferred to Broadway, opening in November 2009, with Catherine Zeta-Jones as Desiree Armfeldt and Angela Lansbury as Madame Armfeldt. Other members of the original London cast also transferred with the production. The production closed in January 2011 after 425 performances.

In 2010, Nunn directed a revival of the Andrew Lloyd Webber musical Aspects of Love from July to September 2010 at the Menier Chocolate Factory and the play Birdsong, which opened in September 2010 at the Comedy Theatre, based on the Sebastian Faulks novel of the same title.

As of May 2011, Nunn was in a relationship with Nancy Dell'Olio, but by the end of the year it was reported the relationship had ended.

In 2014, Nunn told the Telegraph that Shakespeare was his religion. “Shakespeare has more wisdom and insight about our lives, about how to live and how not to live, how to forgive and how to understand our fellow creatures, than any religious tract. One hundred times more than the Bible. I’m sorry to say that. But over and over again in the plays there is an understanding of the human condition that doesn’t exist in religious books.”