The Belgian, 36, who left his job as Anderlecht head coach last month, said he was "excited by the challenge ahead".
"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.
Jackson replaced Sean Dyche in April but could not save the club from relegation to the second-tier Championship.
"I'm excited by the challenge ahead," said Kompany.
"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."
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