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Kompany, 36, has agreed to take charge at Turf Moor after parting ways with Anderlecht.
Kompany left the Belgian club by mutual consent last month after a third-place finish in the Jupiler Pro League.
He had been in charge of his boyhood club since 2020, winning 43 of his 95 games as head coach.
The former City captain has now chosen to return to English football with Burnley, who were relegated from the Premier League on the final day of the season.
He becomes the permanent successor to Sean Dyche, who was sacked by the Clarets in April after 10 years as manager.
"Burnley Football Club is a truly historic English side and it is an honour to be appointed first-team manager," said Kompany.
"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."
A City legend, Kompany first joined the Manchester club from Hamburg in 2008 and would play a key role in the team's transformation under the ownership of Sheikh Mansour.
Kompany would go on to make 359 appearances for City, captaining the club for eight seasons and winning four Premier League titles, two FA Cups and four Carabao Cups during his time at the Etihad.
He left City in 2019 after 11 seasons at the club to become player-coach at Anderlecht.
He later relinquished his managerial duties in order to focus on his playing career, before taking over full-time in 2020 after announcing his retirement from professional football.
Kompany guided Anderlecht to a fourth-place finish in the Belgian league in his first season in charge and could only manage a third-place finish this term, finishing behind surprise champions Royale Union Saint-Gilloise and second-place Club Brugge.
He is now tasked with securing Burnley an immediate return to the Premier League.
Featured Image Credit: PA Images