The son of John Crimp, a British Rail signalling engineer, and his wife Jennie, Crimp's family moved in 1960 to Streatham where he attended a local primary school before winning a scholarship to Dulwich College. But when his father was transferred to York, he went to the nearby Pocklington School, where he showed an aptitude for languages, music, English literature and theatre. He read English at St Catharine's College, Cambridge 1975–78, where his first play Clang was staged by fellow student Roger Michell. Before establishing himself as a playwright, he put together An Anatomy, a collection of short stories, and also wrote a novel Still Early Days. These remain unpublished.
His first six plays were performed at the Orange Tree Theatre in Richmond. As he told Marsha Hanlon in an interview for the Orange Tree appeal brochure in 1991: "When the Orange Tree ran a workshop for local writers [in September 1981], I was invited to take part. The carrot was the chance of a lunchtime production, so I wrote Living Remains and the Orange Tree staged it — my first-ever produced play! I was so excited that I didn't think about the space where it was performed [then a room above a pub], but now I realise that the Orange Tree's intimacy and simplicity provided an extra layer of excitement."
Seven of his plays, and his second Ionesco translation have also been presented at the Royal Court Theatre, London, where he became writer-in-residence in 1997. His plays are now frequently performed in Europe. He has also been an assiduous translator of European texts.
Possibly his most highly regarded, and certainly his boldest and most innovative play is Attempts on Her Life, first performed at the Royal Court in 1997 and subsequently translated into twenty languages.
In this work, none of the lines are assigned to a particular character, nor does Crimp specify how many actors should perform the piece. In seventeen apparently disconnected scenes, groups of people give mutually contradictory descriptions of an absent protagonist, a woman talked of as if she were, variously, a terrorist, the daughter of grieving parents, an artist and a new car. This deliberately fragmented work challenges an audience to re-define its notion of what constitutes a "play" and might seem to question whether someone has any existence beyond the models we construct.