Born in London, Curzon studied at the Royal Academy of Music. His public debut was at a Prom in 1923 when he played a Bach triple concerto under Henry Wood. Between 1928 and 1930 he took further instruction from Artur Schnabel in Berlin. then studied under Wanda Landowska and Nadia Boulanger in Paris. He toured throughout Europe and the United States. He was knighted in 1977.
Curzon was particularly well known for his interpretations of Wolfgang Amadeus Mozart and Franz Schubert. In his earlier years he had been noted for his championing of modern music - Lennox Berkeley's Piano Sonata is dedicated to him.