Kapoor had been suffering for some time from kidney disease and had been admitted to hospital last week.
He belonged to a famous family of filmmakers and actors, and was a prominent star in 1950s and 1960s.
One of the most popular actors of his generation, Kapoor starred in hits like Junglee, An Evening in Paris, Chinatown and Kashmir Ki Kali.
Bollywood superstar Amitabh Bachchan tweeted that with Kapoor's death the "flamboyance and joie de vivre of the [film] industry [was] lost".
Shammi Kapoor acted in more than 100 films. Fans called him the "Elvis Presley of India" for his frenetic and nifty dancing.
He developed a style of his own and moviegoers flocked to the theatres just to see Shammi Kapoor's antics and mannerisms.
With his infectious on-screen persona and energy, Shammi Kapoor was Bollywood's first real star, say correspondents.
'Yahoo!' The actor was also a keen internet buff and amongst the first Indians to have a website of his own.
When Yahoo opened its office in Mumbai several years ago he was invited by its co-founder Jerry Yang.
At the launch, Kapoor was pleasantly surprised to hear the band playing his Yahoo song from the film Junglee [or Wild], made famous years before the internet existed, and so called for his famous cry of "Yahoo!".
Later, Mr Yang told him how inspired he was by the Yahoo song and the way the actor had used the word in his inimitable style in so many of his films.
"It was all very flattering. Many of my relatives still call up and ask whether I own Yahoo," he told the BBC in an interview.