The polymath first found fame in the early 1960s in the revue Beyond The Fringe with Peter Cook, Dudley Moore and Alan Bennett after giving up a career as a doctor, and went on to become one of the most influential figures in British culture of his generation.
He made his directing debut in 1962 with John Osborne's Under Plain Cover and went on to direct theatre and television plays, including The Merchant Of Venice at the National Theatre and six of the BBC Television Shakespeare plays.
His started directing opera in the 1970s, when he worked for Kent Opera, Glyndebourne and English National Opera, with whom he had a four-decade relationship, while his Royal Opera production of Cosi fan tutte was a staple of the repertory for nearly 20 years.
His family said in a statement: "Our father died this morning peacefully at home with his family around him following a long battle with Alzheimer's.
"He was 85 and leaves his wife Rachel and children Tom, William and Kate. His death is a great loss to our family and to his friends and will leave a huge hole in our lives."
Sir Jonathan was also acclaimed as an author - writing books on subjects ranging from Freud to acting in opera - an artist, photographer, sculptor and television presenter, fronting shows such as The Body In Question and Atheism: a Rough History Of Disbelief.
He was also an artistic director of the Old Vic and was knighted in 2002 for services to music and the arts.
Tony Hall, director-general of the BBC, said: "Jonathan Miller was a creative genius whose imagination knew no bounds.
"He was a doctor; a writer; producer; director; presenter - a creator who enjoyed a remarkable career in theatre, television and opera. He brought arts and culture to millions on the BBC."
Oliver Mears, director of opera at the Royal Opera House, said: "Jonathan Miller was one of the most important figures in British theatre and opera of the past half century.
"Combining a supreme intellect with a consistently irreverent perspective, formed from his experiences in both comedy and medicine, Miller shone a unique light on our art form.
"His intolerance of inauthenticity and laziness on stage was matched by the urgency and rigour of his search for the composer's vision, historical accuracy and psychological truth - resulting in so many productions which have stood the test of time.
"As artistic director of the Old Vic, he also gave some of our most brilliant practitioners their first chance - a legacy that lives on today. He will be sorely missed."