Downing, who arrived at Villa's training ground on crutches because of a broken foot that will keep him out of action until September, completed a medical and signed a four-year-deal.
The left-sided midfielder, who turns 25 next week and has 22 England caps, said his desire to play in next year's World Cup finals was a major factor behind his move after Middlesbrough's relegation from the top flight last season.
"Leaving Boro feels strange because I've always been at the club. I supported them when I was young and I'll always be a fan," Downing, who joined Middlesbrough as a trainee, told the club's website.
"But I hope the fans will understand that I'm ambitious and want to win things. This move gives me a chance of that.
"Also I definitely want to play at next year's World Cup, so that's my aim - to get back to full fitness, get back playing and get myself in the England team for the World Cup.
"I don't just want to go there (South Africa) to sit it out this time either. I want to be in the team. I loved the last World Cup, it was brilliant, but I want to play a bigger part this time.
"I have to be with a good Premier League side if I want to do that," he added.
Boro manager Gareth Southgate, who rejected Downing's transfer request in January when Tottenham Hotspur made an offer, said:
"I felt that for the last year or so Stewart has been looking to move to another club and I think it was impossible for us to stand in his way at this time.
"From my point of view, I have loved working with Stewart. He has played very well for his home-town club."
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