The 22-year-old Scotland international has signed a five-year contract at the Emirates, and will compete with Sead Kolasinac for the left-back spot.
He follows Nicolas Pepe, William Saliba and Gabriel Martinelli as Arsenal's fourth permanent signing of the transfer window.
“We’re delighted Kieran is joining us," said manager Unai Emery. "He’s a very talented player who will continue to improve. He increases our options defensively and I look forward to him joining our group.”
Lifelong Celtic fan Tierney joined the club at the age of seven, going on to make 170 appearances after making his first-team debut in 2015. The club released a statement saying that they "did all we could" to keep him at Celtic Park.
“This is an opportunity Kieran wanted to take and while we will be sad to see him leave the club, we wish Kieran all the very best for the future," manager Neil Lennon told the club's website.
“Although he is still a young guy, Kieran has given the club so much. He’s a great lad and I’ve really enjoyed working with him. I’m sure he will continue to progress and develop and make a big impact in the English Premiership.
“It has obviously been a long process, but now it has been finalised we can now look ahead and concentrate on continuing to build our own squad and meet the many challenges ahead, domestically and in Europe."
The deal to take Tierney to north London has dragged on throughout the summer, with issues over the structuring of the payment having caused issues to the point it had seemed the move would break down entirely.
However, it seems that Tierney's desire to test himself at a higher level has played a part in pushing the deal through.
Inevitably, he has drawn comparisons with Liverpool left-back and international team-mate Andy Robertson, and Gunners fans will be hoping he can make a similar impact on the flank.
Pending the final registration details, Tierney should be available for Arsenal's first game of the new Premier League season, as they travel to Newcastle on Sunday.