The British physicist was known for his work with black holes and relativity, and wrote several popular science books including A Brief History of Time.
"We are deeply saddened that our beloved father passed away today," a family statement said.
At the age of 22 Stephen Hawking was given only a few years to live after being diagnosed with a rare form of motor neurone disease.
Obituary: Stephen Hawking
The illness left him wheelchair-bound and largely unable to speak except through a voice synthesiser.
His family said that he died peacefully in his home near Cambridge University, where he did much of his ground-breaking work on black holes.
In the statement his children, Lucy, Robert and Tim, said: "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 humour" 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."
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.
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 neurone disease and given two years to live
• Outlined his theory that black holes emit "Hawking radiation" in 1974
• Published his book A Brief History of Time in 1988, which has sold more than 10 million copies
• His life story was the subject of the 2014 film The Theory of Everything, starring Eddie Redmayne
The scientist gained popularity outside the academic world and appeared in several TV shows including The Simpsons, Red Dwarf and The Big Bang Theory.