The 26-year-old Germany international had long been linked with a move to The Emirates and both clubs finally announced that an agreement had been reached on Monday.
Podolski, who has signed a “long-term” contract with the Gunners, said: “I’m so happy to be joining Arsenal Football Club and to play in the Premier League.
“Arsenal is one of the top clubs in Europe with a huge history. There are many top quality players at Arsenal and the style of football which the team plays is fantastic.
“I’m proud to become an Arsenal player and am looking forward to playing my first match at Emirates Stadium and doing my best for all the Arsenal fans.”
He added: “I am looking forward to becoming an Arsenal player, but at the moment, my focus is with my current club Cologne.
“I will give everything I have until the end of season to make sure the team finish as high as possible in the Bundesliga. There will always be a place in my heart for Cologne.”
Arsenal boss Arsène Wenger said that he sees Podolski as an “important” part of the London club’s future.
He said: “We are delighted to secure the deal for Lukas and see him as an important part of our future. He is a top-class player, a very good finisher and a proven performer at club and international level. He is a very strong player and will provide us with good attacking options.
“We are happy to have made this signing early and we are looking forward to watching Lukas contribute at the European championships over the summer, a level where he has already made 95 caps at the age of 26. That is a phenomenal record and just shows his quality as a player.”
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