The former England international has signed a deal until 2019.
Richards, 26, will officially join Villa on a free transfer on July 1, having been released by City after spending last season on loan at Serie A club Fiorentina.
Reports last week claimed that the ex-England international had turned down lucrative offers from Sunderland, Inter Milan and West Ham United following discussions with manager Tim Sherwood, with the Villa boss believed to have earmarked him as a replacement at centre-back for soon-to-be free agent Ron Vlaar [h/t Guardian].
Speaking to the club’s official, Sherwood said: "I'm really pleased to get Micah on board and I'm sure he will prove to be an excellent signing for the football club."
City’s longest-serving player at the time of his departure, Richards made 245 appearances for the Blues after earning his first-team debut in October 2005, aged just 17, but struggled for game time in recent seasons after being usurped at both right-back and centre-half.
Joining Fiorentina on a season-long loan deal in September, he featured 19 times for La Viola but, having failed to claim a regular place in the line-up, the Italian outfit opted against making his deal permanent for a reported £4.7 million fee.
Hailed as a future England stalwart upon making his international bow in November 2006, the 13-times capped defender will now bid to get his once-flourishing Premier League career back on track at Villa, where he is believed to be earning around £65,000-per-week.