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Mitchell was born near Hampstead Heath, in North London. His mother, Kathleen Fabian, worked at the Friedrich Froebel Children's Educational Institution, and his father Jack Mitchell was a chemist. Adrian staged his first play at the age of nine, while still at school. After graduation, Mitchell served in the Royal Air Force, where his "natural pacifism" manifested itself. He went on to study English at Christ Church College, Oxford, where his teacher was the son of J. R. R. Tolkien.
He became chairman of the University Poetry Society and literary editor of Isis magazine. After graduation, he got a job as a reporter in the Oxford Mail newspaper, and later in the London Evening Standard. As the editor of the poetry section in the New Statesman magazine, he was the first to publish an interview with The Beatles.

