The Sylvester Medal is a bronze medal awarded by the Royal Society (London) for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian Professor of Geometry at the University of Oxford in the 1880s, and first awarded in 1901, having been suggested by a group of Sylvester's friends after his death in 1897. Initially awarded every three years with a prize of around £900, the Royal Society have announced that starting in 2009 it will be awarded every two years instead, and is to be aimed at 'early to mid career stage scientist' rather than an established mathematician. The award winner is chosen by the Society's A-side awards committee, which handles physical rather than biological science awards.

James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the American Journal of Mathematics. At his death, he was a professor at Oxford University.