The former Liverpool boss takes over on a 12-month rolling contract.
Brendan Rodgers is the new Celtic manager.
The Carnlough man, sacked by Liverpool in October 2015, replaces Ronny Deila as the new boss at Parkhead.
“I am absolutely delighted to be named Celtic manager,” he said.
“This is genuinely a huge honour for me. I have followed Celtic all my life and to be given this fantastic opportunity and to be part of such a truly great football club is a dream come true.
“I will give my new role everything I have and do all I can to bring our supporters exciting, entertaining and winning football.
“The Club has been in magnificent shape in recent years and has collected silverware regularly during this time. My objective now, of course, is to continue this work, to keep us at the top and again make our mark in Europe.
“I know what a magnificent support Celtic enjoys and I can promise our fans one thing – that I will be doing everything I can to give you a team that you can be proud of and a team that delivers.
“I can’t wait to be in Paradise with our team and our fans as we all get to work.”
Celtic Chairman Ian Bankier commented: “We are delighted to welcome Brendan to Celtic as the Club’s new manager.
“As expected, there was huge interest in the position of Celtic manager but we believe we have chosen the outstanding candidate to take the Club forward.”
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