The actor was well known for playing Bilbo Baggins in The Hobbit and The Lord Of The Rings trilogies.
His agent, Alex Iriwn, confirmed he “died peacefully” in hospital on Friday morning.
Alex said Ian’s illness was “Parkinson’s related”.
In a statement, he added: “[Ian] was a genius of stage and screen, winning multiple awards and loved by directors, audiences and his colleagues alike.
“His sparkling wit always accompanied a mischievous twinkle in his eye.
“Charming, kind and ferociously talented, we will miss him hugely”.
Ian Holm was born on 12 September 1931 in Goodmayes, Essex.
Inspired by seeing a production of Les Misérables as a boy, he secured a place at the Royal Academy of Dramatic Art in 1949, before joining the Royal Shakespeare Company.
His role as Sam Mussabini in Chariots Of Fire earned him a special award at the Cannes Film Festival, a Bafta award and an Oscar nomination for Best Supporting Actor.
He also found a new audience in the 1990s in the role of Pod in the TV adaptation of The Borrowers.
Sir Ian’s other screen credits included The Fifth Element, Alien, The Sweet Hereafter, Time Bandits, The Emperor’s New Clothes and The Madness Of King George.
He was also a celebrated theatre star, winning critical acclaim for his role as King Lear at the National Theatre in 1998, as well as a Tony Award for Best Featured Actor as Lenny in The Homecoming.
Sir Ian was married four times, most recently to Sophie de Stempel, and had five children.