The former Manchester City defender left Belgium side Anderlecht back in May after three years in charge.
Burnley are back in the second tier of English football following their relegation at the end of the 2021/2022 campaign.
The Clarets sacked long-serving boss Sean Dyche in April but his interim replacement Mike Jackson could not help them retain their top flight status.
"Burnley is a truly historic English side and it is an honour to be appointed first-team manager," Kompany said following his appointment.
"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 added: "We are absolutely delighted to welcome Vincent to Burnley.
"Vincent is a proven leader and I've been very impressed with his ideas for Burnley, 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."
As a player, Kompany joined Man City from Hamburg in 2008 and starred for eleven seasons, making 323 appearances and captaining the side to four Premier League titles, two FA Cups and four League Cups.
