Khan was known for his nuanced and understated performances, with many calling him one of India's most talented actors.
He was so highly respected, director Wes Anderson once wrote him a part just so he could work with him.
In 2018, the actor revealed he had a neuroendocrine tumour, a rare cancer.
He later underwent treatment in a London hospital for the illness, which affects cells that release hormones into the bloodstream.
Khan had been in intensive care in a Mumbai hospital since Tuesday with a colon infection.
Bollywood stars and politicians rushed to pay tribute on social media.
"An incredible talent... a gracious colleague... a prolific contributor to the World of Cinema... left us too soon... creating a huge vacuum," superstar actor Amitabh Bachchan who worked with him on the film Piku, tweeted.
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