The star had been suffering from cancer, a statement said.
He became one of Britain’s best-loved acting stars thanks to roles including Professor Snape in the Harry Potter films and Hans Gruber in Die Hard.
Harry Potter author JK Rowling led the tributes, describing him as “a magnificent actor and a wonderful man”.
She wrote on Twitter: “There are no words to express how shocked and devastated I am to hear of Alan Rickman’s death.”
She added: “My thoughts are with [Rickman’s wife] Rima and the rest of Alan’s family. We have all lost a great talent. They have lost part of their hearts.”