The news was announced on his official Facebook page, but no details about the cause of death were given.
“It is with profound sorrow we report that legendary poet, songwriter and artist, Leonard Cohen has passed away,” the statement said.
Canadian Prime Minister Justin Trudeau led tributes to the singer, who was known for hits including Hallelujah and Dance Me to the End of Love.
“It is with deep sorrow that I learned today of the death of the legendary Leonard Cohen,” Mr Trudeau said in a statement.
“He will be fondly remembered for his gruff vocals, his self-deprecating humour and the haunting lyrics that made his songs the perennial favourite of so many generations.
Fans have begun gathering outside Cohen’s Montreal home to light candles and lay flowers in memory of the singer.
Cohen’s son Adam told Rolling Stone: “My father passed away peacefully at his home in Los Angeles with the knowledge that he had completed what he felt was one of his greatest records.
“He was writing up until his last moments with his unique brand of humour.”
A memorial for Cohen will take place in Los Angeles at a later date, the Facebook announcement added.
The Montreal-born singer’s hits included Suzanne, Bird on the Wire and I’m Your Man and he released his 14th album, You Want It Darker, just last month.
He was inducted into the Rock and Roll Hall of Fame in 2008.
“This is a very unlikely occasion for me. It is not a distinction that I coveted or even dared dream about,” he said.
Born into a Jewish family, Cohen later became a student of Zen Buddhism.
For five years, from 1994 to 1999, he took a break from the music industry and lived at Mount Baldy Zen Center, east of Los Angeles.
“My life was filled with great disorder, with chaos, and I achieved a little discipline there,” he said. “So I decided to return to music.”
Cohen was made a Companion of the Order of Canada in 2003.
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