The 36-year-old spent three years as Anderlecht boss, but has quit to take over the recently relegated club.
Kompany will be tasked with getting Burnley promoted back to the Premier League.
Last season, he led the Belgium side to third place, securing European qualification, and also took them to a domestic cup final.
As a player, he was hugely successful with Man City and captained them to four Premier League titles, two FA Cups, and four League Cups.
"Burnley Football Club is a truly historic English side and it is an honour to be appointed first-team manager," said Kompany, who takes over from caretaker Mike Jackson who replaced Sean Dyche in April.
"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."
Burnley chairman, Alan Pace said: "We are absolutely delighted to welcome Vincent to Burnley Football Club.
"Vincent is a proven leader and I've been very impressed with his ideas for Burnley Football Club, his appetite to succeed and his focus on leading the club back to the Premier League.
"Vincent has shown impressive credentials in leading one of Belgium's biggest teams back to European football and a Cup Final last season and we've been excited by his philosophy, approach and ambition for the club."