In 1959, he won first prize in the Pablo Casals Competition in Mexico. Later he was the principal cellist for six years in the Royal Concertgebouw Orchestra from 1962 to 1968. He became the Erasmus Scholar at Harvard University in 1982. He is the author of the book Bach, the Fencing Master, a stylistic and aesthetic analysis of Bach's Cello Suites. He is one of the pioneers of the 'Dutch Baroque School' and rose to fame as a partner of Frans Bruggen and Gustav Leonhardt, who toured extensively together and made many recordings. Bylsma continues to be a towering figure in the baroque cello movement.
In 1979 Bijlsma recorded the first ever recording of the six Suites for unaccompanied cello (BWV 1007-1012) by J. S. Bach on a period instrument. He also made a second recording of the same music in 1992 on the large "Servais" Stradivarius and on a five-string violoncello piccolo.
Anner Bylsma is married to Dutch violinist Vera Beths.