In 1987, Kempf won the first National Mozart Competition in England[8] and in 1992, was named BBC Young Musician of the Year for his performance of Rachmaninoff's Rhapsody on a Theme of Paganini.

[9] Kempf's early adult career benefited from his failure to win the 1998 International Tchaikovsky Competition in Moscow, where the first prize in the piano section went instead to Denis Matsuev.

Apparently, some judges had wanted to award the first prize jointly to Matsuev and Kempf, and had successfully negotiated with the Russian Culture Ministry for the additional funding.

However, Kempf only collected the third prize in the end, which provoked a barrage of indignant protests from both the audience and the Russian press, who accused some of the judges of bias, especially towards contestants who also happened to be their former pupils.

[10] Kempf has continued to perform solo, chamber and concertante music in Europe, the Americas, East Asia and Australia, and has recorded recital discs of Bach, Beethoven, Chopin, Liszt, Prokofiev, Rachmaninoff and Schumann.

"[13] Bridle went on to note, "His craving for chamber music came from the Marlboro Festival", and cited Kempf recalling that, "I remember being told I'd have to rehearse the Mozart Piano Trio on the first day.