  Let R be a commutative ring and let n,m be two positive integers. The
symmetric group on n letters acts diagonally on the ring of polynomials in nxm
variables with coefficients in R. The subrings of invariants for this action is
called the ring of multisymmetric functions since these are the usual symmetric
functions when m=1. In this paper we will give a presentation in terms of
generators and relations that holds for any R and any n,m answering in this way
to a classical question. I would like to thank M.Brion, C.De Concini and
C.Procesi, in alphabetical order, for useful discussions.
