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The 50-year-old is best known for his spell at Basel, who were bottom of the table when he arrived in October 2023. The following season, they finished as champions and won the Swiss Cup.
Celestini has spent a decade in management, also leading the likes of Lausanne-Sport, Luganbo, Luzern and Sion.
As a player, he won 35 caps for Switzerland and featured against England at the Euros in 2004.
Celestini has been out of work since leaving CSKA Moscow - where he won the Russian Super Cup – at the end of last season.
He replaces Nicky Hayen, who was sacked following the 4-1 defeat at Lincoln City over the weekend with the Clarets sitting in the relegation zone.
The new head coach will be in the dugout against Portsmouth at Turf Moor this weekend.
“I look at Burnley and I see a club with big potential and players with real quality, who can help us achieve our supporters’ ambitions," he said.
"I know the situation we are in, and I have lived it before. Our supporters deserve to see a team playing with belief from the first whistle.”
Chief football officer Diego Lopez added: “Fabio brings the passion, experience and knowledge we were looking for.
"He showed extraordinary determination and conviction to come to Burnley. We want people who truly want to be here and help this club grow. Fabio is a clear example of that.”