The 29-year-old has signed a three-year deal with the Scottish Premiership champions following five years at Aberdeen.
Celtic midfielder Ryan Christie has joined Aberdeen on a season-long loan as part of the deal that took Hayes to Celtic Park.
“This is a huge honour for me. It is a dream come true to be part of this brilliant football club and I can’t wait to get started,” Hayes told the club’s website.
Hayes, whose last goal for Aberdeen came in last month’s Scottish Cup final defeat to Celtic, is already relishing the opportunity to test himself in next season’s Champions League.
“I can’t wait to get back training,” Hayes said. “Once everything was signed off, I said to Tim Williamson (Celtic physio) that I was just looking forward to that first day in training and getting started.
“You can’t get away from the Champions League. I can imagine it is the reason many players have wanted to come here. Not only is it the size and stature of the club, Champions League nights and the atmosphere are a massive factor in wanting to come here.
“Whether you’re a football fan or not, everybody knows about the atmosphere of Champions League nights at Celtic Park. I’d be lying if I said I didn’t want a part of that.”
Hayes began his career in Reading’s academy but he never made a senior appearance for the Royals and signed for Leicester in 2007.
He struggled for game time at the Foxes and spent time on loan at Northampton and Macclesfield before joining Inverness Caledonian Thistle on a free transfer in 2009.
After three years in the Highlands he moved east to Aberdeen, for whom he has made over 200 appearances and scored 30 goals.