Norris grew up in Africa and Malaysia, attended North Bromsgrove High School and Kidderminster College of Further Education, and later trained as an actor at RADA before turning to directing.

In 2001 he won the Evening Standard Award for Outstanding Newcomer for his production of David Rudkin's Afore Night Come at the Young Vic.

From 2002 to 2007 Norris was an Associate Director at the Young Vic, where his productions have included Feast by Yunior Garcia Aguilera, Rotimi Babatunde, Marcos Barbosa, Tanya Barfield and Gbolahan Obisesan (2013), Vernon God Little by DBC Pierre (2007), adapted by Tanya Ronder (2009 and 2011), Hergé's Adventures of Tintin, which Norris adapted, with David Greig (Barbican, 2005; UK tour & West End, 2007) and his own adaptation of Sleeping Beauty (Young Vic, 2002; Barbican, 2004, UK & international tour).

His production of the Kander and Ebb musical Cabaret, produced by Bill Kenwright, ran at the Lyric Theatre in London's West End from 2006 to 2008.

[4] He created the 2011 production Dr Dee for the Manchester International Festival, in collaboration with musician Damon Albarn, which was subsequently performed at ENO in 2012.

His production of London Road by Alecky Blythe and Adam Cork opened at the Cottesloe in 2011, before transferring to the Olivier in 2012, winning the Critics' Circle Award for Best Musical.

[7] In summer 2016 he directed Bertolt Brecht and Kurt Weill's The Threepenny Opera, in a new adaptation by Simon Stephens and starring Rory Kinnear as Macheath.