The 33-year-old retired from rugby in 2007 after being plagued by a serious neck injury but has since returned to the game and the England squad.
And the 2003 World Cup winner, whose current side Leeds sit on the verge of relegation from the Aviva Premiership, could not hide his delight at signing with the Adams Park outfit.
"I'm excited about the chance to play with Wasps for the next few seasons - I've been really impressed with some of their young players," said the 66-cap man.
"I'm looking forward to being part of a team that is aiming to get back to the top of their game."
"As much as it has been a difficult season for Leeds on the pitch, I have really enjoyed my time there and met some great people.
"I am very grateful that they gave me the opportunity to return to the English game."
Wasps chief executive Chris Thomas said Thompson was "exactly the type of front-row forward we have been looking for in our recruitment drive for next season".
"He will bring with him excellent experience at international and Premiership level and we believe that Steve will also play a big role in developing some of our future talent," said Thomas.