Forman passed away on Friday in the United States, where he lived, after a short illness, his wife, Martina, told Czech news agency CTK on Saturday.
"His departure was calm and he was surrounded the whole time by his family and his closest friends," she said.
Forman was born in the Czech town of Caslav on Feb. 18, 1932. He moved to the United States after the Communist crackdown on the "Prague Spring" uprising in 1968 and became a U.S. citizen in the 1970s.
"One Flew Over the Cuckoo's Nest," in which a psychiatric institution becomes a microcosm of the contemporary world, and "Amadeus," the life of 18th-century composer Wolfgang Amadeus Mozart through the eyes of his rival Antonio Salieri, earned 13 Oscars between them, including those for best director to Forman.
His other notable work included the rock musical "Hair" in 1979, "Ragtime" in 1981 and "The People vs Larry Flynt" in 1996, which was nominated for an Academy Award that year.
Jim Carrey, star of Forman's 1999 Andy Kaufman biopic "Man on the Moon", said on Twitter: "Another great one passes through the doorway. Milos Foreman. What a force. A lovely man. I’m glad we got to play together. It was a monumental experience."
Larry Karaszewski, the co-writer for "Man on the Moon" and "The People vs Larry Flynt", called Forman a master filmmaker.
"No one better at capturing small unrepeatable moments of human behavior," he said on Twitter. "Milos loved life. I will miss his laughter."