Cheng, who had been diagnosed with a rare illness with symptoms similar to Parkinson's disease, died on Wednesday at home surrounded by her loved ones, her family wrote on Facebook.
They said medical treatment could not slow the progression of the disease, called neurodegenerative atypical parkinsonism syndrome, and she had donated her brain for medical research.
"Our mom wanted to be remembered by how she was: the legendary Queen of Martial Arts ... a versatile, award-winning actress whose film and television career spanned over six decades, not only in Asia but internationally as well," they said.
The entertainment publication Variety reported that she died in the San Francisco Bay Area.
The Shanghai-born film star became a household name in Hong Kong, once dubbed the Hollywood of the Far East, for her performances in martial arts movies in the 1960s.
One of her most famous films was Come Drink with Me, directed by King Hu and released in 1966.
She played Jade Fox, who uses poisoned needles, in Crouching Tiger, Hidden Dragon, which also starred famed Hong Kong actor Chow Yun-fat and Oscar-winning Michelle Yeoh.
The movie, released in 2000, won four Oscars.
More recently, Cheng performed in Disney's Mulan, released in 2020.
She is survived by her three daughters and a son.
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