Northern Irishman Rodgers, who has also managed at Swansea City, Reading and Watford, was sacked by Liverpool in October and replaces Ronny Deila at the Scottish club.
“I am absolutely delighted to be named Celtic manager,” Rodgers, 43, said in a statement on Celtic's website (www.celticfc.net).
“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.”
One of the most pressing tasks facing Rodgers will be steering the club into the group stage of the Champions League, starting with a second qualifying round tie.
He will also face the renewal of the 'Old Firm' rivalry with their Glasgow neighbours Rangers, who will return to the top flight next season after four years in the lower divisions following the club's liquidation in 2012.
“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,” Celtic Chief Executive Peter Lawwell said.
“We have appointed a special manager and we are sure he can bring some special times to Celtic.”
Rodgers was sacked by Liverpool after an underwhelming start to the season, with his record in the transfer market coming under scrutiny and his team struggling in the Premier League.
Having been named Liverpool boss in 2012, Rodgers took the club to the brink of the Premier League title two years later, playing an exhilarating brand of football, spearheaded by striker Luis Suarez.
A late slump that season, however, allowed Manchester City to pip them to top spot and from that point on things went downhill for Rodgers.
Liverpool sold Suarez to Barcelona and brought in several players with the proceeds, with many failing to make an impact as the club slumped to sixth in the table.
A humiliating 6-1 defeat at Stoke City in Liverpool's final game of the 2014-15 season left Rodgers on borrowed time and he was dismissed after eight league games of the following campaign.
Original source: Brendan Rodgers named manager of Celtic