Boris Johnson said he "inspired the lives of countless young people".
"It is with deep sorrow that Her Majesty The Queen announces the death of her beloved husband," the palace said.
"The Royal Family join with people around the world in mourning his loss."
Speaking at Downing Street, the prime minister said that the duke had "earned the affection of generations here in the United Kingdom, across the Commonwealth, and around the world".
Meanwhile, Justin Welby, the Archbishop of Canterbury, said he "consistently put the interests of others ahead of his own and, in so doing, provided an outstanding example of Christian service".