Stephen Hawking died peacefully at his home in Cambridge in the early hours of Wednesday, his family said.
The British scientist was famed for his work with black holes and relativity and wrote several popular science books including A Brief History of Time.
At the age of 22, Prof Hawking was given only a few years to live after being diagnosed with a rare form of motor neuron disease.
The illness left him in a wheelchair and largely unable to speak except through a voice synthesizer.
In a statement his children, Lucy, Robert, and Tim said: "We are deeply saddened that our beloved father passed away today.
"He was a great scientist and an extraordinary man whose work and legacy will live on for many years."
They praised his "courage and persistence" and said his "brilliance and humor" inspired people across the world.
"He once said, 'It would not be much of a universe if it wasn't home to the people you love.' We will miss him forever."
A book of condolence has been opened at Gonville and Caius College in Cambridge, where Prof Hawking was a fellow.
Prof Hawking was the first to set out a theory of cosmology as a union of relativity and quantum mechanics.
He also discovered that black holes leak energy and fade to nothing - a phenomenon that would later become known as Hawking radiation.Through his work with mathematician Sir Roger Penrose, he demonstrated that Einstein's general theory of relativity implies space and time would have a beginning in the Big Bang and an end in black holes.
The scientist gained popularity outside the academic world and appeared in several TV shows including The Simpsons, Red Dwarf and The Big Bang Theory.
His most famous book - A Brief History of Time - has now shot to the top of the Amazon Best Sellers list.
The Motor Neurone Disease Association, of which Prof Hawking had been a patron since 2008, reported that its website had crashed because of an influx of donations to the charity.
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