The defensive midfielder featured in all five of his country's World Cup matches in Russian Federation and becomes Unai Emery's fourth summer signing, joining Stephen Lichtsteiner, Bernd Leno and Sokratis Papastathopoulos.
Torreira played a part in all of Uruguay's World Cup matches and started their quarterfinal defeat to France last Friday.
Torreira, 22, told Arsenal's official website: "We give all we have to our football, our jersey and our country".
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At the Veterans of Foreign Wars national convention in Kansas City on Tuesday, he said, "I think the process is starting very soon, I hope".
"Sokratis and Lichtsteiner's age and experience balanced off with one (maybe two) exciting young players. We welcome Lucas to Arsenal and look forward to him joining us soon for pre-season", he added.
His biggest deal to date is to acquire Lucas Torreira, a budding superstar who impressed at the World Cup for Uruguay.
Arsenal have completed the signing of Uruguay global Lucas Torreira from Sampdoria. I hope I can do great things here.
"I feel calm", Torreira said.
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But Roglič sounds happy about the upcoming test against the clock. Geraint Thomas takes a two-minute lead into the time trial . I'm not bothered about Tom Dumoulin or Chris Froome .
"I want to win titles, titles, titles".
Click through the gallery above to see Arsenal's signings in this window so far.
Torreira's signing has shifted expectations among fans heading into a new era that a top-four finish is now the minimum requirement in Emery's first season in charge. Starting his youth career in Uruguay with I.A 18 Julio and then Montevideo Wanderers, Lucas moved to Italy in 2013 where he joined Pescara.
Press Association Sport understands that Arsenal are also close to completing a deal to land France Under-20 worldwide Matteo Guendouzi.
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She said they didn't talk business, however, and he asked whether she was single before asking to kiss her. But the magazine story hadn't actually come out until shortly before 3 p.m.
And Emery was delighted to have signed the 19-year-old.