The former England international has signed a three-year deal in Scotland after falling down the pecking order at Spurs.
New Tottenham head coach Nuno Espirito Santo has brought in Pierluigi Gollini on loan, with the Italian is set to compete with captain Hugo Lloris for the No1 spot this season.
“This is a great moment for me in my career and I am absolutely delighted to be joining a club of Celtic’s stature,” said Hart after joining Celtic.
“I know just how big this club is and of the incredible support that it enjoys, and I can’t wait to play in front of the fans at Celtic Park.
“I’m looking forward to meeting up with my new team-mates now and getting to work with the squad and, in particular, the other goalkeepers here.”
Celtic have also signed midfielder James McCarthy on a free transfer after his contract at Crystal Palace expired at the end of last season.
Of the 11 players who were out of contract at Palace, only Christian Benteke and Joel Ward have been given new deals.
McCarthy said: “To have signed for Celtic is a brilliant feeling and this is a special day for me and my family.
“I know all about the club, the size of it and all the success it’s enjoyed, particularly in recent years. I’m looking forward to pulling on the Hoops, playing at Paradise and helping the team deliver more success to these amazing fans in the seasons ahead.”