Gautier Capuçon was born in Chambéry, Savoie, the youngest of three siblings. His brother is the violinist Renaud Capuçon.
He started learning the cello when he was four years old. He began his formal musical education in his hometown at the Ecole Nationale de Musique de Chambéry, where he graduated with first prizes in cello and in piano.
In Paris, he studied the cello initially with Annie Cochet-Zakine, who had heard him in Chambéry and brought him with her to the Conservatoire Supérieur de Paris (CNR), where he graduated in 1997 with the first prize in cello. He then became a pupil of cello pedagogue Philippe Muller at the Conservatoire National Supérieur de Musique de Paris (CNSMP), where he graduated in 2000 with first prizes in cello and chamber music. After that, he finished his studies with Heinrich Schiff at the University of Music and Performing Arts, Vienna.
Between 1997 and 1998, as a student, he was a cellist in the European Community Youth Orchestra (now the European Union Youth Orchestra) and also in the Gustav Mahler Jugendorchester, playing under conductors including Bernard Haitink, Pierre Boulez and Claudio Abbado.
He is also an accomplished pianist. He started learning the piano at the age of seven and studied it as his second instrument at the Conservatoire Supérieur de Paris under Christophe Egiziano. He enjoys playing jazz piano recreationally. Now he teaches at Classe d'Excellence de Fondation Louis Vuitton with young students.