He died peacefully at his home in Cambridge in the early hours of Wednesday, his family said.
The Briton was known 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 neurone disease.
The illness left him in a wheelchair and largely unable to speak except through a voice synthesiser.
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 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.”
A book of condolence is due to be opened at Gonville and Caius College in Cambridge, where Prof Hawking was a fellow.
On being diagnosed with motor neurone disease: ‘My expectations were reduced to zero when I was 21. Everything since then has been a bonus’ – Interview in The New York Times, December 2004.
On black holes: ‘Einstein was wrong when he said, ‘God does not play dice’. Consideration of black holes suggests, not only that God does play dice, but that he sometimes confuses us by throwing them where they can’t be seen’ – The Nature Of Space And Time, published 1996.
On God: ‘It is not necessary to invoke God to light the blue touch paper and set the universe going’ – The Grand Design, published 2010.
“I regard the brain as a computer which will stop working when its components fail. There is no heaven or afterlife for broken-down computers; that is a fairy story for people afraid of the dark.”
“I believe the simplest explanation is, there is no God. No one created the universe and no one directs our fate. This leads me to a profound realisation that there probably is no heaven and no afterlife either. We have this one life to appreciate the grand design of the universe and for that, I am extremely grateful.”
“I have lived with the prospect of an early death for the last 49 years. I’m not afraid of death, but I’m in no hurry to die. I have so much I want to do first.”
“We are just an advanced breed of monkeys on a minor planet of a very average star. But we can understand the Universe. That makes us something very special.”