Born in London, Sylvester studied mathematics at Cambridge but was not granted his BA degree since he was a practicing Jew. The relevant statute was later revoked and Sylvester was granted both his BA and MA in 1871. He was widely read in a number of languages and was a keen amateur musician and a prolific poet. Feeling unable to keep an academic post as a mathematician Sylvester worked first in an insurance company and later as a lawyer. In 1876 he went to America to become the first professor of mathematics at Johns Hopkins University. He became the first editor of the American Journal of Mathematics and did much to develop mathematics in America. He returned to England in 1883 to become Savilian Professor of Geometry at Oxford University.

Sylvester's best mathematical work was in the theory of invariants and number theory. With his lifelong friend the British mathematician Arthur Cayley, he was one of the creators of the theory of algebraic invariants, which proved to be of great importance for mathematical physics.