New York Mayor Michael Bloomberg said: "America has lost a genius who will be remembered with Edison and Einstein."
Jobs was a college dropout when he co-founded Apple in a garage 35 years ago and his vertiginous rise to success saw him become a billionaire business icon and he left a company with a bigger cash pile than the United States government.
He was the creative genius who gave the world the iPod - the product that transformed popular culture and revolutionised the music business.
Product launch events for his gadgets - which also included the iPhone, the iPad, and the iMac - were dominated by his exuberant personality but in recent years he had appeared increasingly gaunt as illness took its toll.
Jobs had survived pancreatic cancer in 2004 and a liver transplant in 2009. He had already announced in January that he was taking sick leave for a third time.
In a letter to company directors and the "Apple community" in August announcing he was stepping down, he said: "I have always said if there ever came a day when I could no longer meet my duties and expectations as Apple's CEO, I would be the first to let you know. Unfortunately, that day has come."
Resigning as chief executive, Jobs said: "I believe Apple's brightest and most innovative days are ahead of it."
It was left to successor Tim Cook to roll out the latest iPhone model this week.
Jobs was seen as rescuing the company when he returned as boss in 1997, launching the iMac, whose sleek design made its products cool again.
Born in San Francisco, he founded Apple with Steve Wozniak, five years his senior, in 1976 and eight years later, the Mac home computer, with its unique WIMP interface - windows, icons, mouse and pointer - was born.
But in 1985 Jobs was ousted in a boardroom coup and went on to form a new high-end technology company, NeXT. The firm was bought by Apple and Jobs was soon back at the helm of the company he had started.
Today Apple is the world's second biggest company, valued at US$349 billion ($454 billion), a figure largely built on Jobs's gift for innovation.
Where Bill Gates, his great rival, represented the All-American nerd, Jobs was a college drop-out.
As Robert Cringely wrote in Accidental Empires: "Gates sees the PC as a way of changing the world."
Jobs once said: "You can't just ask customers what they want and then try to give that to them. By the time you get it built, they'll want something new."
Jobs is survived by his wife Laurene, and their three children, Erin, Reed and Eve. Agencies
