Mr Landau won a Best Supporting Actor Oscar in 1995 for portraying the horror movie star Bela Lugosi in Ed Wood.
He died on Saturday in Los Angeles of "unexpected complications" following a hospital visit.
Mr Landau was born in New York and started out as a cartoonist for the New York Daily News before moving to theatre and then cinema acting.
He featured in the Alfred Hitchcock film North by North-West and played a commander in Space-1999 and Geppetto in a live-action version of The Adventures of Pinocchio.
But he turned down the role of Mr Spock in Star Trek, a role that went to his friend Leonard Nimoy instead.
And Mr Nimoy later replaced Mr Landau on Mission: Impossible when the latter left following a dispute over pay.
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