  In this paper we introduce poly-Poisson structures as a higher-order
extension of Poisson structures. It is shown that any poly-Poisson structure is
endowed with a polysymplectic foliation. It is also proved that if a Lie group
acts polysymplectically on a polysymplectic manifold then, under certain
regularity conditions, the reduced space is a poly-Poisson manifold. In
addition, some interesting examples of poly-Poisson manifolds are discussed.
