Rodgers came close to masterminding Liverpool’s first English title since 1990 in 2013/14, only for their challenge to slip away.
Celtic chief executive Peter Lawwell described Rodgers as “a special manager.”
He said: “We wanted to bring one of the biggest and best names to the club to match our own aspirations and those of our supporters – we believe, in appointing Brendan, that we have done this,” he said.
“We have appointed a special manager and we are sure he can bring some special times to Celtic.
“Brendan is a highly-sought after manager and we are pleased that we have been able to bring such a high calibre individual to Celtic.
“He has a huge and genuine affection for the club.
I know Brendan feels privileged to be named Celtic manager and I know he will give us everything he has to be successful at Celtic.”