Kantorow was born in Cannes, France. From the age of 13 he studied at the Paris Conservatoire with René Benedetti, and in 1960 won the first violin prize. In the 1960s he won ten major international prizes, including first prizes in the Carl Flesch Competition (London), the (GenoaPaganini Competition, and the Geneva International Competition. Since the 1970s he has been noted for his solo performances in a very wide range of repertoire (from Baroque music to contemporary), and as a chamber music performer. His recordings have won many awards, including the Grand Prix du Disque and the Grand Prix de l’Académie Franz Liszt. He held senior positions at the Strasbourg and Rotterdam conservatories and at the Conservatoire de Paris, until his retirement from conservatoire violin pedagogy. He continues to teach privately and to give master-classes.

In the 1980s he began a separate career as conductor, becoming principal conductor of the Auvergne Chamber Orchestra and later the Ensemble Orchestral de Paris. He has longstanding conducting engagements with other European orchestras, including the Netherlands Chamber Orchestra, the Tapiola Sinfonietta of Finland, the Helsinki Chamber Orchestra, and the Lausanne Chamber Orchestra. From 2004 to 2008 he was principal director of the Orquesta Ciudad de Granada in Spain.