After sacking Sean Dyche in April, Burnley appointed Mike Jackson as interim boss, but he was unable to keep the Lancashire club in the Premier League.
Ahead of the upcoming Championship campaign and a potential promotion push, the Clarets have confirmed Kompany as their new manager.
The Belgian, who won the Premier League as a player on four occasions, takes over at Turf Moor after spending two years in charge of Anderlecht.
Following his appointment, Kompany told the Burnley FC website: "Burnley Football Club is a truly historic English side and it is an honour to be appointed first-team manager. I'm excited by the challenge ahead.
"I'm looking forward to getting to work with the players and creating a positive, winning team for our fans when we return to Turf Moor.
"I've been impressed by the Board's vision for the club which aligns with my own and I look forward to playing my part as we enter an important season."
The length of Kompany's contract has not been specified by Burnley.
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