The 29-year-old will arrive at Vitality Stadium on 1st July from Premier League champions Chelsea for an undisclosed fee, having put pen to paper on a long-term deal.
Begovic began his career at Portsmouth, and enjoyed a brief loan spell with the Cherries in 2007, making eight appearances before returning to Fratton Park.
The commanding stopper joined Stoke City in February 2010, quickly establishing himself as the Potters’ number one.
After more than 170 appearances for Stoke, Begovic joined Chelsea in the summer of 2015 and has represented the Blues in the Premier League, Champions League, FA Cup and EFL Cup.
To date, he has won 56 caps for Bosnia and Herzegovina, featuring at the 2014 World Cup and being named the 2012 Bosnian Footballer of the Year.
“AFC Bournemouth is an exciting, ambitious club which has everything in place for a very bright future,” said Begovic.
“Being part of a title-winning team at Chelsea was amazing but I felt now was the right time to move on because I want to be playing regular football.
“I am coming into an important part of my career and I want to show people what I am capable of.
“When I spoke to Eddie Howe I knew AFC Bournemouth was the right club for me and I want to bring my experience to the squad, both on and off the pitch.
“I’m really excited to be part of the journey here and hope I can help take the club to another level.”
AFC Bournemouth manager Eddie Howe said: “Asmir is an outstanding goalkeeper who has proved himself capable of performing at the highest level over a number of years.
“He is at the right age and the right fit. He is a leader as well and exactly the kind of character you want in the team.
“Asmir will provide great competition to the excellent group of goalkeepers we already have here and his experience will prove invaluable in the dressing room and on the pitch.