Jobs introduced the iPod and the iPhone to the world
Apple said his "brilliance, passion and energy were the source of countless innovations that enrich and improve all of our lives. The world is immeasurably better because of Steve".
Jobs announced he was suffering from pancreatic cancer in 2004.
US President Barack Obama said that with his death, the world had "lost a visionary".
Microsoft boss Bill Gates said Jobs' "profound impact" on the world of technology would "be felt for many generations to come"."Steve was among the greatest of American innovators - brave enough to think differently, bold enough to believe he could change the world, and talented enough to do it," said Mr Obama.
Mr Gates added: "For those of us lucky enough to get to work with him, it's been an insanely great honour. I will miss Steve immensely."
And New York Mayor Michael Bloomberg said that "America lost a genius who will be remembered with Edison and Einstein, and whose ideas will shape the world for generations to come".'The face of Apple'
In the statement, Apple said Jobs's "brilliance, passion and energy were the source of countless innovations that enrich and improve all of our lives".
Life of Steve Jobs
"The world is immeasurably better because of Steve."
A message on the Apple website says they have "lost a visionary and creative genius, and the world has lost an amazing human being".
Flags are being flown at half mast outside the Apple headquarters in Cupertino, California, while fans of company have held vigils and left tributes outside Apple shops around the world.
"What he's done for us as a culture, it resonates uniquely in every person," said Cory Moll, an employee at an Apple shop in San Francisco.
"Even if they never use an Apple product, the impact they have had is so far-reaching."
At the company's Shanghai shop, customer Jin Yi said Jobs had created gadgets which had "changed people's perceptions of machines".
Jobs had built a reputation as a forthright and demanding leader who could take niche technologies - such as the mouse and the graphical window-based interface - and make them popular with the general public.
He introduced the colourful iMac computer, the iPod and the iPhone to the world. His death came just a day after Apple unveiled its latest iPhone 4S model. Tributes have been left at Apple's headquarters and its shops around the world
In 2004, Jobs announced that he was suffering from pancreatic cancer and he had a liver transplant five years later.
In January, he took medical leave, before resigning as CEO in August and handing over his duties to Tim Cook.
In his resignation letter, Jobs said: "I believe Apple's brightest and most innovative days are ahead of it. And I look forward to watching and contributing to its success in a new role."
However, Jobs stayed on as Apple's chairman.
More than almost any other business leader, he was indistinguishable from his company, which he co-founded in the 1970s.
Apple - whose market value is estimated at $351bn (£227bn) - is now the world's most valuable technology company. Only oil giant Exxon Mobil is worth more.
As the face of Apple, Jobs represented its dedication to high-end technology and fashionable design.
And inside the company he exerted a level of influence unheard of in most businesses.
Despite a high profile, however, he remained fiercely protective of his private life.
He married his wife Laurene in 1991, and the couple had three children.
Jobs also leaves a daughter from a previous relationship, and as an adult he discovered that he had a biological sister, US novelist Mona Simpson.
Steve Job, thanks for all that you did for us.
We love you and shall continue to be loyal to Apple always.
Editor Social Journalist.