In 2005, Owen was nominated for an Academy Award and won a Golden Globe and BAFTA for Best Supporting Actor for his role in the film version of Closer.

Owen was born in Coventry, West Midlands, England, the fourth of five brothers. At the age of three, he was abandoned by his father, a country and western singer. He was raised by his mother and step-father. Initially opposed to drama school, he changed his mind in 1984, after a long and fruitless period of searching for work. He graduated from RADA in 1987 in a class including Ralph Fiennes and Jane Horrocks. After graduation, he won a spot at the Young Vic, performing in several Shakespeare plays. In an incident he later described as "very schmaltzy", he met his future wife Sarah-Jane Fenton while they were performing the title roles of Romeo and Juliet. They married in 1995, and have two daughters, Hannah and Eve. They currently live in London.

The 1990s saw him become a regular on stage and television in the UK, notably his lead role in the TV series Chancer and in the West End and Broadway hit play Closer, by Patrick Marber. Owen won critical acclaim for his performance in the 1997 Mike Hodges film Croupier, where he played the title role of a struggling writer who takes a job in a London casino as inspiration for his work, only to get caught up in a robbery scheme. He also starred in the well-written British mystery series Second Sight, in which he played DCI Ross Tanner. He became well known to North American audiences after starring as The Driver in the BMW films.

Despite public denials, Owen had long been rumoured to be a possible successor to Pierce Brosnan in the role of James Bond. Rumours intensified after Brosnan admitted that he would not play the part again, but in October of 2005 it was announced that British actor Daniel Craig would take Brosnan's place. Unconfirmed sources stated that Owen had turned down the role to avoid typecasting. In 2006 he spoofed the Bond connection by making an appearance in the remake of The Pink Panther in which he plays a character named "Nigel Boswell, Agent 006" (another character in the film quips that Owen's character is "one short of the big time", referring to James Bond, who is widely recognized as Agent 007).