Born in Johannesburg, she is a niece of anti-apartheid campaigner, Helen Suzman. Suzman tends to run towards classical works, performing in many productions of Shakespeare plays, and in such works as The Three Sisters and Hedda Gabler. Her marriage to director Trevor Nunn, which ended in divorce, was a famous theatrical alliance. She appeared in many British television drama productions in the 1960s and early 1970s. Her first film role was in 1971, in Nicholas and Alexandra, and she was nominated for the Academy Award for Best Actress for her portrayal of the Empress Alexandra. She has made few films since, the best-known being Nuns on the Run (1990), a rare comedy performance.