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 acquired a local following in Manchester, and were signed 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 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, just a month shy of turning twenty years old, joined AC/DC after their drummer, Phil Rudd, left in May 1983. 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 left the group in November 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.
In 2005, Wright participated on a tribute album to Heavy metal icons Iron Maiden. It was his second of three times doing a tribute disc; the first experience being in 1998 on an AC/DC tribute album titled Thunderbolt, while in 2013 he took part in the DIO Tribute album "This Is Your Life".
In 2006 Wright is credited in the song "Lucy in the Sky with Diamonds" on the album Butchering the Beatles, a heavy metal tribute.