From 2012 - 2015, he was Director, Entertainment Channels at Sky overseeing Sky 1, Sky Living, Sky Arts and the launch of Sky Atlantic. Sky won its first Emmy Awards and Oscar nominations as well as BAFTAs, British Comedy Awards and Royal Television Society Awards during this period. Murphy also had responsibility for Sky Arts, which as well as a channel is an on-demand library of arts and cultural content in Europe. 
In 2003 Murphy launched BBC Three and commissioned Little Britain, Gavin and Stacey and Flashmob The Opera. Before BBC Three, he was Channel Controller of BBC Choice, and previously he ran UK Play, a music and comedy channel co-owned by BBC Worldwide.
Under Murphy the ENO aims to broaden the appeal of opera and mix up the age, ethnicity, and social background of the audience at the ENO.
As a teenager, he played clarinet for the Leeds Youth Orchestra and Leeds Youth Opera.
Stuart grew up in Leeds, where his passion for classical music was sparked by playing the clarinet in Leeds Youth Orchestra and Leeds Youth Opera.
He attended St. Mary's Menston in West Yorkshire. He studied Political Geography at Clare College, Cambridge in 1990. He has two sons with his ex-wife. He is openly gay and first spoke publicly about his sexual orientation in a 2012 interview.
His partner, David Clews is Creative Director of TwoFour, and directed the BAFTA-award winning Educating Essex.
He started his career as a tea boy at BBC Manchester working in DEF II programming. He then worked on shows such as Reportage, The Sunday Show, Fist of fun, and Lifeswaps with Paul O'Grady. He later worked as a producer at MTV on Hanging Out with Davina McCall, and at the Big Breakfast. He re-joined the BBC to work for Jane Root in the Independent Commissioning Group, and later developed Radio One TV for Roly Keating, on UKTV.
He launched and ran UK Play, a music and comedy channel owned by UKTV from 1998.
Became the first channel controller of BBC Three, which launched in February 2003. He commissioned comedies including Little Britain (which had been originally commissioned by BBC Radio 4), Pulling, and Early Doors as well as various dramas including Torchwood, and Conviction. He kickstarted parenting programming on TV, with Who Rules The Roost, Honey, We're Killing The Kids, Little Angels and The House of Tiny Tearaways both presented by Tanya Byron. Other commissioned shows included Flashmob The Opera (a live opera from Paddington station) and Flashmob The Opera: Meadowhall.
He was tipped as an outsider in the running for the controllership of BBC One in 2007. In 2004 The Observer included Murphy in a list of 80 young people who they believed would shape people's lives in the early 21st Century.
Between 2006 and 2008, he was the Creative Director of Twofour Broadcast.
He joined Sky1 in May 2009 and commissioned a variety of drama, entertainment, and factual programmes including Got to Dance, Must Be the Music, A League of Their Own, Terry Pratchett's Going Postal, Strike Back, Mad Dogs, The Runaway, Little Crackers, Ross Kemp: Middle East Special, Pineapple Dance Studios, Louie Spence's Showbusiness, An Idiot Abroad, Trollied, Mount Pleasant, Spy, Stella, Starlings, among many others.
In November 2010 he was made Director of Commissioning across all Sky Entertainment channels. He was also given responsibility for launching Sky Atlantic, a new entertainment channel which is the home of the majority of HBO content in the UK. He bought Mad Men (previously at the BBC), Entourage (previously at ITV) as well as Blue Bloods and The Borgias. He commissioned screenwriter Paul Abbott to make Sky Atlantic's first drama, Hit & Miss, starring Chloë Sevigny, commissioned cult comedy This is Jinsy, brought Alan Partridge back to TV, commissioned Kathy Burke's Walking and Talking, Julia Davis's new comedy Hunderby, as well as documentary Flying Monsters with David Attenborough.
Sky Atlantic launched on 1 February 2011.
In May 2012, he was made Director of Entertainment Channels at Sky, overseeing all of the entertainment and commissioning portfolio.
In November 2015, he made the decision to leave Sky to pursue more creative endeavours, he has signed a deal to develop his own scripts with Sky Vision.
In March 2018, Stuart was appointed Chief Executive of the English National Opera, where he joins Daniel Kramer and Martyn Brabbins on ENO's leadership team.