Peter, 55, is the 12th actor to play the Doctor in the long-running science fiction series.
He will take over from Matt Smith, who at 26 became the youngest galaxy-hopping Time Lord who travels in the Tardis, a time machine shaped like an old-fashioned British police telephone booth.
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A Doctor Who fan since childhood, Peter said that "being asked to play the Doctor is an amazing privilege."
"Like the Doctor himself I find myself in a state of utter terror and delight," he said. The role will be a big change for the Glasgow-born actor, best known for playing spin doctor Malcolm Tucker in comedy series The Thick of It.
"I think Malcolm has been banished from the mirror by this Doctor Who, who certainly would not put up with any of Malcolm's language or attitude," said the actor on the BBC show on Sunday.
Peter is the oldest actor to play the Time Lord since the first Doctor, William Hartnell, in 1963.
He was most recently seen starring alongside Brad Pitt in the zombie epic World War Z, and has previously won an Oscar for directing Franz Kafka's It's a Wonderful Life.
Doctor Who marks its 50th anniversary on 23 November with a special episode, starring Matt Smith, David Tennant and John Hurt as the Doctor.
Matt, who recently hung up his tweed jacket from playing the Time Lord, said in his official leaving statement: "Thank you so very much for supporting my incarnation of the Time Lord, number 11, who I might add is not done yet, I'm back for the 50th anniversary and the Christmas special!"
"It's been an honour to play this part, to follow the legacy of brilliant actors, and helm the Tardis for a spell with 'the ginger, the nose and the impossible one'. But when ya gotta go, ya gotta go and Trenzalore calls. Thank you guys."