Goalkeeper, Begovic, who is 28, has signed a four-year contract, and he becomes the team’s second signing of the summer, following the arrival of Radamel Falcao on a season-long loan.
Begovic told Chelsea website, “I am very happy to be joining Chelsea FC. After speaking at length to the manager, I feel like I can develop here and be an important part of this team. I am looking forward to meeting up with the team on Wednesday for the pre-season tour.”
Begovic had spent the previous five-and-a-half years with Stoke, having joined from his first professional club, Portsmouth, in early 2010.
Playing with Stoke City at the Britannia Stadium, he made 172 appearances, quickly establishing himself as first choice. He scored the Premier League’s longest-ever goal when he netted a clearance from inside his own penalty area against Southampton in November 2013.
Born in the former Yugoslavia, Begovic spent his childhood in Germany and Canada, representing the latter at the Under-20 World Cup before opting to play at senior level for Bosnia-Herzegovina.
He was their first-choice goalkeeper at their 2014 World Cup finals debut in Brazil, and has accumulated 41 caps so far.
Former Chelsea goalkeeper, Petr Cech joined Arsenal on a long-term contract for an undisclosed fee in June.
Begovic takes over the vacant number one shirt while Thibaut Courtois will retain his number 13 shirt.
Chelsea has also announced that Radamel Falcao will wear the number 9 shirt during the 2015/16 campaign.