Lambert joins from Norwich City, having offered his resignation earlier in the week, after leading them to a very respectable 12th place on their return to the Premier League last season.
Villa dismissed Alex McLeish, also a Scot, the day after the season ended last month.
McLeish, never popular with the fans after arriving from arch-rivals Birmingham City, ended his one season in charge with the former European champions in 16th place and just two points off the relegation zone.
They won just seven of their 38 league games.
The club announced the news on their website with a one line statement: "The Board of Aston Villa are delighted to confirm that Paul Lambert has been appointed Villa manager."
The signing of 42-year-old Lambert, a former midfielder who won the Champions League with Germany's Borussia Dortmund in 1997 and also played for Celtic, follows immediately after Liverpool's swoop for Swansea City boss Brendan Rodgers.
Swansea and Norwich, both small provincial clubs, were promoted together and both impressed last season with their young British managers catching the eye of the American owners at the two former European Cup winners.
Villa's American chairman Randy Lerner said last month that the Birmingham-based club lacked the sort of "compelling play" the fans expected.
Lambert, a player under successful former Villa manager Martin O'Neill in his time at Celtic, will be expected to deliver as he becomes the fourth man in charge at Villa in less than two years.