Wright began his career with local band Tora Tora (not to be confused with the American glam metal group), before joining Manchester group A II Z, a New Wave of British Heavy Metal band founded in 1979 in Manchester, England by guitarist Gary Owens. The full line-up consisted of David Owens (vocals), Gary Owens (guitar), Gam Campbell (bass), Karl Reti (drums). They rapidly acquired a local following in Manchester, and were quickly snapped up by Polydor Records, eager to cash in on the exploding NWOBHM boom. A single live album, The Witch of Berkley, followed in 1980. Reti was subsequently replaced by Wright. The band then promptly disintegrated and Wright went on to perform in Tytan and recorded on their full-length debut album Rough Justice, however; that band also dissolved shortly before the release of the debut.
Wright joined AC/DC after their drummer, Phil Rudd, was fired in 1983 (reinstated in 1994). AC/DC recorded three albums with Wright in the mid-late 80s; Fly on the Wall, Who Made Who and Blow Up Your Video. Wright then left the group in 1989 to join Dio, and was replaced by Chris Slade. Simon Wright joined Rhino Bucket when he wasn't working with Dio to replace the departed Liam Jason for their third album, 1994's independently released Pain. In 2007, he reunited with the band and is featured on their latest album The Hardest Town.
Wright has had two stints with Dio, 1990–91 and 1998–2010. With the band he has recorded four studio albums (Lock up the Wolves, Magica, Killing the Dragon and Master of the Moon) and two live albums (Evil or Divine - Live In New York City and Holy Diver - Live).
His drumming can also be heard on albums by UFO, Michael Schenker Group and John Norum.
In 2005, Wright participated on a tribute album to Heavy metal icons Iron Maiden. It was his second time doing a tribute disc; the first experience being in 1998 on an AC/DC tribute album titled Thunderbolt.
In 2006 Wright is credited in the song "Lucy in the Sky with Diamonds" on the album Butchering the Beatles, a heavy metal tribute.
In August 2009, Wright teamed up with Joe Lynn Turner, Phil Soussan and Carlos Cavazo as part of the Big Noize project, playing shows in Iraq and Kuwait