The 22-year-old defensive midfielder has completed a move from Sampdoria.
The Gunners confirmed the move on their official website. Torreira is seen as a key player for new manager Unai Emery, who wants someone physical and energetic in central midfield.
In a statement to the club's official website, Emery said: "In Lucas Torreira, we have signed a young player who is a very bright talent in the game. A midfielder with great quality, I have enjoyed watching his performances for Sampdoria in the past two seasons, and we all saw him do very well for Uruguay in the World Cup.
"He is a young player already with good experience, but who wants to keep on growing. We welcome Lucas to Arsenal and look forward to him joining us soon for pre-season."
Torreira will wear the number 11 shirt at the club. The deal is subject to him obtaining a work permit, which is seen as a formality.