Maxwell's parents are Welsh (his mother acted in the premiere of Dylan Thomas's Under Milk Wood both in the West End and on Broadway) but he was born and grew up in Welwyn Garden City in Hertfordshire. He studied English at Worcester College, Oxford. He began an MLitt there, but in 1987 moved to America to study poetry and drama with Derek Walcott at Boston University.

His three earliest collections of poetry, Tale Of The Mayor's Son (1990), Out of the Rain (1992), Rest For The Wicked (1995) are collected as The Boys at Twilight: Poems 1990-1995 (2000).

1994 he was named one of the New Generation poets and he received the E. M. Forster Award in 1997. His book Time's Fool (2000) is a narrative poem written in terza rima, and is now in development as a film. His most recent collections are The Nerve (2002, winner of the Geoffrey Faber Memorial Prize), The Sugar Mile (2005) and Hide Now, which was published in 2008 and shortlisted for the T.S.Eliot Prize.

His work is included in several anthologies of the best English poetry of the 20th century, as well as in the latest edition of the Norton Anthology of English Poetry.

His first novel Blue Burneau (1994) was shortlisted for the Whitbread First Novel Prize and the book Moon Country, published in 1996, describes a visit to Iceland with Simon Armitage. His second novel The Girl Who Was Going To Die was published in 2008 by Cape in the UK and by Kunstmann in Germany.

His play Liberty, about the French Revolution, premiered at Shakespeare's Globe in the 2008 season (dir. Guy Retallack), prior to a UK tour. Also that year, The Only Girl in the World was revived at the Arcola (dir. Alex Clifton); Mimi and the Stalker premiered at theatre503, produced by Giudecca, (dir. Michael Gieleta); The Lifeblood was revived in New York, produced by Phoenix Theatre Ensemble (dir. Bob Hupp) at the Connolly Theater, and The Ruins, a version of Hecabe/The Trojan Women, premiered at RADA (dir.Alex Clifton).

His radio adaptation of Dostoevsky's The Gambler will be broadcast on BBC Radio 3 (on Sunday 14 June 2009), starring Patricia Routledge, Sam Crane, Siobhan Hewlett and Nicholas Le Prevost, and directed by Guy Retallack. This is also in development as a stage play.

His new play The Frontman, about the circumstances surrounding a murder in a school, premieres at the Underbelly on the Edinburgh Fringe this summer (dir. Jon Croker).

The Lifeblood, concerning the last days of Mary, Queen of Scots, was British Theatre Guide's 'Play of the Fringe' at Edinburgh in 2004, and was directed by Guy Retallack with Sue Scott Davison as Mary. The Lifeblood was first performed at the Hen and Chickens Theatre in 2001 with Felicity Wren as Mary.

His play Mimi and The Stalker is one of six projects awarded funding by the UK Film Council in the spring 2009 quarter, for development as a screenplay.

Other plays include Wolfpit, about two green children said to have appeared in Suffolk in the 12th century (Edinburgh 1996; New York 2006), The Forever Waltz, a reworking of the Orpheus-Eurydice story (New York 2005; Edinburgh 2005), and The Only Girl in the World, a play about Mary Kelly, the last victim of Jack the Ripper (London 2001). He contributed the fantasy The Black Remote to the National Theatre's Connections series in 2006. He is the Resident Playwright for New York's Phoenix Theatre Ensemble. His earliest plays were staged in the garden of his parents' home in Welwyn in the early 1990s, but were reviewed in the national press.

He has also written opera libretti, including The Girl of Sand, composed by Elena Langer and performed at the Almeida Opera Festival in 2004, and The Birds (after Aristophanes), composed by Edward Dudley Hughes and performed by I Fagiolini at the City of London Festival in 2005. He has two operas in development: The Lion's Face (concerning dementia) with Elena Langer as composer, produced by the Opera Group, premiering at the Brighton Festival in May 2010; and an opera derived from Paradise Lost, composer Luke Bedford, also produced by The Opera Group, scheduled for 2011. A short version of The Lion's Face, (then titled The Present) won the Audience Prize at the Zurich Opera House's New Opera Festival in January 2009.

His verse monologue, The Best Man, was turned into a feature film starring Danny Swanson (dir. Jon Croker).

He has translated several of the early poems of Joseph Brodsky, which will eventually be published by Farrar Straus Giroux in the USA.

He has taught at Amherst College, Columbia University, Princeton University, New York University and The New School in New York City, and for the Poetry School and Goldsmiths College in London. He teaches annual poetry master classes at the Y in New York City. He was Poetry Editor of The New Republic from 2001 to 2007. He reviews for the Times Literary Supplement, London Review of Books and The New Republic. He is a Fellow of the Royal Society of Literature and the Welsh Academy.

He has one daughter, Alfreda (b.1997) and has recently returned to the UK after ten years in the USA. He lives in Islington, London. His daughter (12 present 2009) has done acting classes and movie auditions in London.

He will give the keynote address at the 2010 Griffin Poetry Prize awards ceremony on June 3, 2010.