He won an Oscar in 2012 for the film Beginners and was also nominated for The Last Station in 2010 and All the Money in the World in 2018.
In the latter film he replaced Kevin Spacey, whose complete performance as billionaire J Paul Getty was excised.
His many other films included The Man Who Would Be King and Knives Out.
According to reports, Plummer passed away peacefully at his home in Connecticut with his wife Elaine Taylor at his side.
Lou Pitt, his long-time friend and manager of 46 years, remembered him as "an extraordinary man who deeply loved and respected his profession".
"He was a national treasure who deeply relished his Canadian roots," he continued. "Through his art and humanity, he touched all of our hearts and his legendary life will endure for all generations to come.
"He will forever be with us."
The Academy of Motion Pictures Arts and Sciences, which awards the Oscars, said he "beguiled audiences across generations", adding: "He will be missed." Bafta added their condolences, praising his "amazing work since the 50s".
The official Twitter account for The Sound of Music added they were "saddened" to hear of his death.
Actor Eddie Marsan worked with Plummer on 2016 film The Exception, and said: "It was like watching a master class. He had nothing to prove anymore so he was completely free, kind, funny mischievous and beautiful to watch. RIP."
Plummer had a varied career across film, television and theatre, starring in productions on Broadway and with the Royal Shakespeare Company.
Yet he will be forever known and loved for The Sound of Music, adapted from the Rodgers and Hammerstein musical, in which he appeared alongside Dame Julie Andrews.
Plummer had ambivalent feelings towards his best-known film, which he famously renamed The Sound of Mucus in interviews.
He also likened working with Andrews to "being hit over the head with a big Valentine's Day card, every day", though they later became great friends.
Born Arthur Christopher Orme Plummer in Toronto in December 1929, Plummer grew up in Montreal as an only child and was exposed to the arts by his mother at an early age.
He first studied the piano before devoting himself to acting, having decided that playing the piano professionally "was very lonely and very hard work".
He made his debut on the New York stage role in 1954's The Starcross Story alongside the actress Mary Astor, a show that only ran for one performance.
Yet he soon landed more stage work and later played leading roles at the National Theatre and for the Royal Shakespeare Company.
Plummer made his film debut in 1958's Stage Struck, directed by Sidney Lumet. He was nominated for a Tony the following year and eventually won the award in 1974 for playing Cyrano de Bergerac.
He won his second in 1997 for playing fellow actor John Barrymore in Barrymore.
Plummer was married three times. He and his first wife Tammy Grimes, are the parents of actress Amanda Plummer.