He had returned to India last September after undergoing treatment in the United States for nearly a year, and was admitted to hospital twice in February.
He first appeared on silver screen in 1955 as a child artist in his father's film Shree 420.
Kapoor has acted in more than 90 films and came from a line of actors.
Rishi was a member of the famed Kapoor acting dynasty and was at the pinnacle of his career in the 1970s and 80s.
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Indian prime minister Narendra Modi said Kapoor was a powerhouse of talent: "I will always recall our interactions, even on social media".
Wife Neetu Kapoor was at his side.
He is survived by his wife, Neetu Singh, who co-starred in several of his films, his son Ranbir Kapoor - also a famous Bollywood actor - and his daughter. "He was passionate about films and India's progress".
With India now under lockdown, the Kapoor family issued a statement to urge fans not to publicly gather to pay their respects to Kapoor as is the tradition in the country.
The iconic Bollywood actor had been diagnosed with cancer in 2018. My deepest condolences and strength to the Mr. Kapoor's family, ' added another devastated fan.
"A wonderful actor, with a huge fan following across generations, he will be greatly missed".