Bunting played both piano and the cello as a child, studying with cellist Ivor James at the Royal College of Music. At one point he studied engineering at Bristol University, but eventually left to get a music degree at Cambridge.
After serving in World War II, Bunting studied with cellist Maurice Eisenberg. He made his Wigmore Hall debut in 1952, then traveled to Prades, France, to study with Pablo Casals and play in his festival. Casal's intensive method of preparation had a great influence on Bunting's own teaching and playing.
In 1955, Bunting gave the world premiere of Gerald Finley's Cello Concerto at the Cheltenham Festival; he also premiered Alan Rawsthorne's concerto and performed on the first British broadcasts of Shostakovich's First Cello Concerto and Henze's Ode to the West Wind.
His writing includes the two-volume Essay on the Craft of Cello Playing, published in 1982; he also composed several works for the cello. He was awarded an MBE in 2000.