Lennon, 38, had been in temporary charge of the Glasgow club since they parted company with Tony Mowbray in March.
"Celtic is my home. I've spent the last 10 years of my working life here and I look forward to bringing back the success our great club deserves," the former Northern Ireland midfielder told the team's website.
GEAR:Get your Celtic shirt for new season
Former Celtic and Sweden captain Johan Mjallby, Lennon's assistant last season, told Reuters that he was likely to go back to Parkhead.
"I would expect the club to be in touch in the near future to discuss a deal," said Mjallby. "The main requirement was that Neil would get the manager's job and now he has I would expect Celtic to contact me.
"Celtic are a huge club with supporters all over the world and I fully understand they wanted to take their time in making this appointment. I'm very happy Neil got the job, now he has the chance to bring in his own players and build his own team."
"Neil has Celtic in his blood, he knows the club inside out and is someone who has all the qualities to achieve great things," said chairman John Reid.
"We have been thorough and professional and have made, I am very confident, a decision which will be proved to be a winning one."
