The journalist and former BBC presenter died in his sleep on Friday night.
A statement from his daughters, Samantha and Emma, called him "remarkable", adding: "He had a great life, a wonderful marriage and an enviable career."
Norman hosted BBC One's "Filmâ€¦" show between 1972 and 1998 - its longest running host - as well as writing for the Daily Mail and the Guardian.
His literary agent described him as "the defining voice of film criticism and insightful interviewing of screen legends from both sides of the camera".
Norman's daughters added: "He leaves behind a family who adore him and a great roster of friends who love him too. We will miss him more than we can say."
Norman's literary agent, Gordon Wise, said the presenter had been living with lung cancer for a number of years, but that he would be remembered as "one of the true greats" of film.
"It was probably that background as a newspaper journalist, interviewer and features writer that all came together to make him such a gifted film critic," he told BBC News.
"He could not only appreciate the story of the film but all the talents that had gone into making the film. That is what we all remember from the three decades of the Film programme - that he brought you perspectives from every side of the camera."
Tributes have begun flooding in on Twitter, with plenty of references to Norman's pickled onions - a family recipe handed down from his grandmother that he launched as a range in supermarkets in 2007.
Actor and presenter Stephen Fry tweeted a tribute, writing: "Sad to hear of Barry Norman's departure. A film critic and a provider of fine pickled onions. That's a good life."
Presenter Jonathan Ross, who took over "Film..." in 1999, added to the tributes. He tweeted: "Very sad to hear that Barry Norman has left us. A great critic and a lovely, lovely man."
And Claudia Winkleman, who started presenting the same show in 2010, called him "an incredibly kind man and the greatest critic".
Barry Norman was born in London on 21 August 1933 to film director Leslie Norman and his wife, Elizabeth.
He went to Highgate School in north London, but skipped university, instead beginning his writing career at the Kensington News.
Norman's journalism took him to South Africa to work on two newspapers, but it was when he came back to the UK that his entertainment career took off - becoming showbiz editor at the Daily Mail.
It was during his Fleet Street years that he met his wife Diana - a journalist and the author of best-selling thrillers under her pen name Ariana Franklin - and they wed in 1957.
The couple had two daughters and were married for more than 50 years, until her death in 2011.
Norman became best known as the presenter of the "Film..." programme, which he hosted on BBC One for 26 years, before he left the channel to move to Sky.
His notoriety was solidified when he became a puppet on Spitting Image, which created a catchphrase for him - "And why not?" - that he later adopted himself and which became the title of his autobiography.
He also wrote documentaries on the entertainment industry, wrote columns for the Guardian and the Radio Times, and was the first host of BBC Radio 4's The News Quiz - as well as being a presenter on the Today programme.
BBC director general Tony Hall described Norman as a "first-class presenter and critic".
He added: "Film buffs always found his programmes essential viewing. He dominated broadcasting about films for a generation with wit and great knowledge. He will be greatly missed and our thoughts are with his family and friends."
BBC film critic Mark Kermode described Norman as "the master".
"Watching Barry Norman review films was a pleasure, an education, and an inspiration," he tweeted. "Wit, knowledge and wry enthusiasm."
Comedian Robin Ince also tweeted: "Barry Norman - the creased but alluring portal to Hollywood greats and a lifetime of film fascination."
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Film critic Barry Norman has died aged 83, his family says.