The Scottish actor was best known for his portrayal of James Bond, being the first to bring the role to the big screen and appearing in seven of the spy thrillers.
Sir Sean died peacefully in his sleep in the Bahamas, having been “unwell for some time”, his son said.
His acting career spanned five decades and he won an Oscar in 1988 for his role in The Untouchables.
Sir Sean’s other films included The Hunt for Red October, Highlander, Indiana Jones and the Last Crusade and The Rock.
Jason Connery said his father “had many of his family, who could be in the Bahamas, around him” when he died overnight in Nassau. Much of the Bond film Thunderball had been filmed there.
He said: “We are all working at understanding this huge event as it only happened so recently, even though my dad has been unwell for some time.
“A sad day for all who knew and loved my dad and a sad loss for all people around the world who enjoyed the wonderful gift he had as an actor.”
His publicist Nancy Seltzer said: “There will be a private ceremony followed by a memorial yet to be planned once the virus has ended.”
He leaves his wife Micheline and sons Jason and Stephane.
Daniel Craig, the current James Bond, said Sir Sean was “one of the true greats of cinema”.
“Sir Sean Connery will be remembered as Bond and so much more,” he said.
“He defined an era and a style. The wit and charm he portrayed on screen could be measured in megawatts; he helped create the modern blockbuster.
“He will continue to influence actors and film-makers alike for years to come. My thoughts are with his family and loved ones.”
In reference to Sir Sean’s love of golf, he added: “Wherever he is, I hope there is a golf course.”
Dame Shirley Bassey, who sang the themes to three Bond films including Goldfinger, paid tribute saying: “I’m incredibly saddened to hear of Sean’s passing.
“My thoughts are with his family. He was a wonderful person, a true gentleman and we will be forever connected by Bond.”
Sir Sean, from Fountainbridge in Edinburgh, had his first major film appearance in 1957 British gangster film No Road Back.
He first played James Bond in Dr No in 1962 and went on to appear in five other official films – and the unofficial Never Say Never Again in 1983.
He was largely regarded as being the best actor to have played 007 in the long-running franchise, often being named as such in polls.
Connery made the character of James Bond his own, blending ruthlessness with sardonic wit. Many critics didn’t like it and some of the reviews were scathing. But the public did not agree.
The action scenes, sex and exotic locations were a winning formula.
In truth, his Bond is now a museum piece; the portrayal of women impossibly dated. The action scenes are still thrilling, but the sex too often bordered on the non-consensual.
Thankfully, its been a while since 007 slapped a woman on the backside and forced a kiss. But Connery’s performance was of its time, enjoyed by millions of both sexes and gave the silver screen a 20th Century icon.
Read more about the life of the Scotsman who became a global star
He was knighted by the Queen at Holyrood Palace in 2000. In August, he celebrated his 90th birthday.
Bond producers Michael G Wilson and Barbara Broccoli said they were “devastated by the news” of his death.
They said: “He was and shall always be remembered as the original James Bond whose indelible entrance into cinema history began when he announced those unforgettable words ‘the name’s Bond… James Bond’.
“He revolutionized the world with his gritty and witty portrayal of the sexy and charismatic secret agent. He is undoubtedly largely responsible for the success of the film series and we shall be forever grateful to him.”
Star Wars director George Lucas, who also created the Indiana Jones character, said Sir Sean “left an indelible mark in cinematic history”.
“He will always hold a special place in my heart as Indy’s dad. With an air of intelligent authority and sly sense of comedic mischief, only someone like Sean Connery could render Indiana Jones immediately into boyish regret or relief through a stern fatherly chiding or rejoiceful hug.
“I’m thankful for having had the good fortune to have known and worked with him. My thoughts are with his family.”
Just the seven as 007 – Connery’s Bond movies
Dr No (1962)
From Russia with Love (1963)
You Only Live Twice (1967)
Diamonds are Forever (1971)
Never Say Never Again (1983)
Sir Sean was a long-time supporter of Scottish independence, saying in interviews in the run-up to the 2014 referendum that he might return from his Bahamas home to live in Scotland if it voted to break away from the rest of the UK.
Scotland’s First Minister Nicola Sturgeon said: “I was heartbroken to learn this morning of the passing of Sir Sean Connery. Our nation today mourns one of her best loved sons.
“Sean was born into a working class Edinburgh family and through talent and sheer hard work, became an international film icon and one of the world’s most accomplished actors. Sean will be remembered best as James Bond – the classic 007 – but his roles were many and varied.
“He was a global legend but, first and foremost, a patriotic and proud Scot – his towering presence at the opening of the Scottish Parliament in 1999 showed his love for the country of his birth. Sean was a lifelong advocate of an independent Scotland and those of us who share that belief owe him a great debt of gratitude.”
Alex Salmond, former first minister of Scotland, who was close friends with Sir Sean, described him as “the world’s greatest Scot, the last of the real Hollywood stars, the definitive Bond”.
He said: “Sean Connery was all of these things but much more. He was also a staunch patriot, a deep thinker and outstanding human being.”
He added: “‘Scotland Forever’ wasn’t just tattooed on his forearm but was imprinted on his soul.”