Bunting was born in London in 1924. He began to play piano aged 5, and cello aged 6. He initially studied engineering at Bristol University but switched to studying music at Cambridge in 1946. His studies were interrupted by war service but he completed his BA in 1947. Bunting was a celebrated musician and professor of cello, which he taught at the Royal College of Music. He gave many recitals of cello performances and was awarded an MBE in 2000. He died in 2005 at the age of 80.