Griffiths enjoyed a long career of success not only on film and TV, but also on the stage where he was a Tony-winning character actor.
TV roles included a cookery-loving detective in the BBC series Pie in the Sky.
However, he was best known for playing uncles Vernon Dursley in the Harry Potter series and Monty in Withnail and I (1987).
His Harry Potter co-star Daniel Radcliffe (who also appeared on stage with him in Equus) was among the first to pay tribute, saying: "Richard was by my side during two of the most important moments of my career".
"I was proud to know him. Any room he walked into was made twice as funny and twice as clever just by his presence."
On stage, Griffiths' most acclaimed performance was as the charismatic teacher Hector in Alan Bennett's The History Boys.
After achieving a rare double of winning a Tony Award in New York and an Olivier Award in London, he recreated the role in a 2006 film version.
But it was his role as the predatory Uncle Monty in Bruce Robinson's Withnail And I (one of the biggest cult classics in British cinema history) that made him a fan favourite.
In a message to his co-star on Twitter, the actor Richard E. Grant said: "My beloved Uncle Monty Richard Griffiths died last night. Chin-Chin my dear friend."
Griffiths was born in Thornaby-on-Tees, North Yorkshire in 1947, and left school at the age of 15. He was later persuaded to return to education and studied drama at Stockton and Billingham College in the 1960s before joining the Royal Shakespeare Company.
He married Heather Gibson in 1980 after they met during a production of Lady Windermere's Fan in 1973.
His early TV career saw him land bit parts in series such as Minder, The Sweeney and Bergerac, while he also played small parts in major films such as Chariots of Fire, Superman II and Gandhi before the big breakthrough came in Withnail and I.
Well respected by his peers, he was appointed an OBE in the 2008 New Year Honours for his services to drama.
Sir Nicholas Hytner, director of the National Theatre, said Griffiths's unexpected death would devastate his "army of friends".
He said, "Richard Griffiths wasn't only one of the most loved and recognisable British actors - he was also one of the very greatest.
"His performance in The History Boys was quite overwhelming: a masterpiece of wit, delicacy, mischief and desolation, often simultaneously.
"His anecdotes were legendary. They were, literally, endless. They would go on for hours, apparently without destination, constantly side-splitting."
His agent Simon Beresford described him as a "remarkable man".
He said, "On stage he allowed us to share in our own humanity and constantly question our differences.
"Richard gave acting a good name. He was a remarkable man and one of our greatest and best-loved actors. He will be greatly missed."
Our thoughts go out to his friends and family.