He made his name on stage as a Shakespearean actor and appeared in more than 70 film and TV productions.
He had been suffering from prostate cancer for several years, and died of the disease at his home in Kent.
The Society of London Theatre has announced that the West End will dim its lights at 19:00 BST in the actor’s memory.
Sir Donald’s family described his death as a “huge loss” and asked for their privacy to be respected.
He was appointed a CBE in 1979 and then knighted in 1997 for his services to drama.
Although renowned for his theatre work, he was arguably best known to the masses for his TV appearances – in the sitcom Never the Twain and the BBC legal drama Judge John Deed.