The 19-year-old French left back is the first player in 17 years to join Real from their bitter city rivals Atletico where his brother Lucas, 21, still plays.
"I am thrilled and very happy to be at the best club in the world," he told a news conference.
"I thank the fans for all the support they are giving me. Various teams wanted me but the best option was to go to Madrid to keep growing."
Atletico tried to hang onto the defender, who had an impressive spell on loan at Alaves last season, but he had his heart set on a move to the 12-times European Cup winners.
Asked about possibly lining up against his brother in the Madrid derby, Theo said: "It will be normal, it won't be a war."
Spanish media said Atletico were frustrated by Madrid's signing of one of their most promising players for a transfer fee reported to be 30 million euros ($34.19 million).
($1 = 0.8776 euros)
(Reporting by Rik Sharma; editing by Ken Ferris)
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