Mr McCain's office said he died on Saturday following his battle with brain cancer.
His family had previously said that the one-time presidential candidate would be ending medical treatment.
US president Donald Trump has paid tribute to Mr McCain.
Writing on Twitter, he said: "My deepest sympathies and respect go out to the family of Senator John McCain. Our hearts and prayers are with you!"
Mr McCain's wife Cindy said she was heartbroken by the death of an "incredible man".
In a post on Twitter, she said: "My heart is broken. I am so lucky to have lived the adventure of loving this incredible man for 38 years. He passed the way he lived, on his own terms, surrounded by the people he loved, in the the place he loved best."
Daughter Meghan McCain described her father as a "great fire who burned bright".
She said: "We lived in his light and warmth for so very long. We know that his flame lives on, in each of us."
She added: "His love and his care, ever present, always unfailing, took me from a girl to a woman - and he showed me what it is to be a man."
Barack Obama, who beat Mr McCain in the 2008 presidential election, said the senator had shown what it meant "to put the greater good above our own".
He said: John McCain and I were members of different generations, came from completely different backgrounds, and competed at the highest level of politics.
"But we shared, for all our differences, a fidelity to something higher - the ideals for which generations of Americans and immigrants alike have fought, marched, and sacrificed.
"We saw our political battles, even, as a privilege, something noble, an opportunity to serve as stewards of those high ideals at home, and to advance them around the world.
"We saw this country as a place where anything is possible - and citizenship as our patriotic obligation to ensure it forever remains that way."
Former Secretary of State and 2016 Democratic presidential candidate Hillary Clinton paid tribute to Mr McCain as a "tough politician" and "trusted colleague", saying there will never be another like him.
"@SenJohnMcCain lived a life of service to his country, from his heroism in the Navy to 35 years in Congress. He was a tough politician, a trusted colleague, and there will simply never be another like him. My thoughts and prayers are with Cindy and his entire family," she said on Twitter.
David Cameron hailed Mr McCain as a "great defender of liberty".
The former prime minister said: "The USA has lost a brave, principled and inspiring leader. The world has lost a great defender of liberty.
"RIP Senator John McCain. Prayers and love to your family."
Actor Tom Hanks paid tribute to Mr McCain after his death, saying: "Duty. Honor. Country. Our nation thanks you, John McCain. There has been no finer son of America... Hanx"
And Tesla founder Elon Musk wrote: "The world lost a great man in John McCain. He was an honorable, kind, brave and fundamentally decent human being. If only there were more such people in the world."