The 35-year-old Irishman has signed a five-year contract at Villa Park after the clubs agreed a fee thought to be around £3.5m.
"Aston Villa is a huge club and I'm really excited to be joining. It's a new challenge and I'm eager to embrace it," Given told the club's website.
Since joining City in 2009 the Irish international failed to secure a first team place, losing his first-choice status to England keeper Joe Hart.
"Everybody who has watched him perform over many years in the Premier League will know his pedigree so we believe we have secured the services of one of the best goalkeepers in the business," manager Alex McLeish said of his first signing for Villa.
Given replaces former number one Brad Friedel, who moved to Tottenham Hotspur.