He 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.
— The Big Bang Theory (@bigbangtheory) March 14, 2018
His passing has left an intellectual vacuum in his wake. But it's not empty. Think of it as a kind of vacuum energy permeating the fabric of spacetime that defies measure. Stephen Hawking, RIP 1942-2018. pic.twitter.com/nAanMySqkt
— Neil deGrasse Tyson (@neiltyson) March 14, 2018
Remembering Stephen Hawking, a renowned physicist and ambassador of science. His theories unlocked a universe of possibilities that we & the world are exploring. May you keep flying like superman in microgravity, as you said to astronauts on @Space_Station in 2014 pic.twitter.com/FeR4fd2zZ5
— NASA (@NASA) March 14, 2018
People think it's an interesting coincidence that Stephen Hawking died on Pi Day https://t.co/pl0XlcTgvy
— TIME (@TIME) March 14, 2018
"Look up at the stars and not down at your feet. Try to make sense of what you see, and wonder about what makes the universe exist. Be curious." by Stephen #Hawking is my favourite quote ever.
It really makes me sad that humanity lost this genius man…💔#RIPHawking
— Lina Buschmann (@Khalina1301) March 14, 2018
Stephen Hawking died on Pi Day…..mind blown. So blown.
— FBCW Youth (@fbcw_youth) March 14, 2018
Intelligence is the ability to adapt to change.
-stephen hawking#RIPStevenHawking#RIP #quotestoliveby #quote #quotes #quotesaboutlife #motivationalquotes #motivation #believeinyourself #believe #اقتباسات #إقتباس_من_كتاب #إقتباس #حكمه pic.twitter.com/VzIQ6eHFUN
— Quotes – إقتباسات (@Quote_1002) March 14, 2018
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He was portrayed in both TV and film – recently by Oscar winner Eddie Redmayne in The Theory of Everything, which charted his rise to fame and relationship with his first wife, Jane.
The actor paid tribute to him, saying: “We have lost a truly beautiful mind, an astonishing scientist and the funniest man I have ever had the pleasure to meet.”
Benedict Cumberbatch, who played Prof Hawking in a BBC drama, said he was “a true inspiration for me and for millions around the world”.
His most famous book – A Brief History of Time – has now shot to the top of the Amazon Best Sellers list.
The Motor Neuron 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.
Factfile: Stephen Hawking
Born 8 January 1942 in Oxford, England.
Earned place at Oxford University to read natural science in 1959, before studying for his PhD at Cambridge.
By 1963, was diagnosed with motor neuron disease and given two years to live.
Outlined his theory that black holes emit “Hawking radiation” in 1974.
In 1979, he became the Lucasian Professor of Mathematics at Cambridge – a post once held by Sir Isaac Newton.
Published his book A Brief History of Time in 1988, which has sold more than 10 million copies.
In the late 1990s, he was reportedly offered a knighthood, but 10 years later revealed he had turned it down over issues with the government’s funding for science.
With the Oxford mathematician Roger Penrose, he showed that if there was a Big Bang, it must have started from an infinitely small point – a singularity.
Black holes radiate energy known as Hawking radiation, while gradually losing mass. This is due to quantum effects near the edge of the black hole, a region called the event horizon.
He predicted the existence of mini-black holes at the time of the Big Bang. These black holes would have shed mass until they vanished, potentially ending their lives in an explosion that would release vast amounts of energy.
In the 1970s, Hawking considered whether the particles and light that enter a black hole were ultimately destroyed if the black hole evaporated. Hawking initially thought that this “information” was lost from the Universe. But the US physicist Leonard Susskind disagreed. These ideas became known as the information paradox. In 2004, Hawking conceded that the information must be conserved.