  We define the notion of "diffusion algebras". They are quadratic
Poincare-Birkhoff-Witt (PBW) algebras which are useful in order to find exact
expressions for the probability distributions of stationary states appearing in
one-dimensional stochastic processes with exclusion. One considers processes in
which one has N species, the number of particles of each species being
conserved. All diffusion algebras are obtained. The known examples already used
in applications are special cases in our classification. To help the reader
interested in physical problems, the cases N=3 and 4 are listed separately.
