Jobs, who had battled pancreatic cancer for years, "died peacefully today surrounded by his family," according to a family statement issued Wednesday by Apple.
"We know many of you will mourn with us, and we ask that you respect our privacy during our time of grief," the statement said.
Jobs stepped down as chief executive August 24 but remained chairman of the board until his death.
"Steve's brilliance, passion and energy were the source of countless innovations that enrich and improve all of our lives," Apple's board said in a statement. "The world is immeasurably better because of Steve."
Jobs is credited with introducing not only the mouse and the point-and-click interface in the early days of personal computing but also the more recent touch-screen controls and tablet computers.
He also had a decisive influence on the music industry through Apple's introduction of the iPod digital music player and iTunes online store while the iPhone, which he championed, sparked the smart phone and mobile computing revolutions that have now spread all over the world.
Those successes propelled Apple in recent months to surpass Exxon Mobil Corp as the world's most valuable company with a market capitalization of more than 375 billion dollars.
Jobs' foray into the film world at the helm of the digital animation company Pixar also revolutionized computer graphics with hits like Toy Story and Wall-E.
"Apple has lost a visionary and creative genius, and the world has lost an amazing human being," Tim Cook, who took over from Jobs as chief executive, said in an email to employees.
Calling him an "inspiring mentor," he said Jobs "leaves behind a company that only he could have built, and his spirit will forever be the foundation of Apple."
Cook said Apple was planning "a celebration of Steve's extraordinary life" but gave no details about funeral arrangements.
Tributes poured in from Jobs' rivals and fans alike.
Google Inc's Larry Page called him a "great man with incredible achievements and amazing brilliance" while Facebook Inc's Mark Zuckerberg called him a "mentor and a friend."
"The world rarely sees someone who has had the profound impact Steve has had, the effects of which will be felt for many generations to come," said Microsoft Corp's Bill Gates, perhaps his greatest rival through his storied technology career.
"For those of us lucky enough to get to work with him, it's been an insanely great honour," Gates said. "I will miss Steve immensely."