  We introduce poly-symplectic groupoids, which are natural extensions of
symplectic groupoids to the context of poly-symplectic geometry, and define
poly-Poisson structures as their infinitesimal counterparts. We present
equivalent descriptions of poly-Poisson structures, including one related with
AV-Dirac structures. We also discuss symmetries and reduction in the setting of
poly-symplectic groupoids and poly-Poisson structures, and use our viewpoint to
revisit results and develop new aspects of the theory initiated by D.Iglesias,
J.C Marrero and M. Vaquero.
