Costa leaves Chelsea after three controversial years at the club where he scored 59 goals in 120 appearances to help them win two Premier League titles and one League Cup.
The 28-year-old’s time at Stamford Bridge was ended by Antonio Conte via text message in the summer and in an act of defiance, he stayed in Brazil and refused to return to England.
But Costa has now got his wish to return to the club he left three years ago.
Costa said: “I am very happy to return to my home, I have always said, Atletico is my home, I am very very very happy.
“I have a lot of desire to start contributing. I will give everything for the team, as I always try to do.”
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Costa, who will wear the number 18 shirt, will train with his new team-mates on Wednesday but is unable to play until January 1 due to Atletico’s transfer ban.
Ahead of Chelsea’s meeting with Atletico on Wednesday in Champions League, Conte stressed he has no problem with Costa.
“As I said before, I have no problem with Diego Costa,” Conte said.
“I said that we wish, for him, the best for the future.
“We want to thank him for the last season. But, I repeat, I have no problem with him.”