In this paper we classify the irreducible symplectic surfaces, i.e., compact, connected complex surfaces with canonical singularities that have a holomorphic symplectic form \(\sigma \) on the smooth locus, and for which every finite quasi-étale covering has the algebra of reflexive forms spanned by the reflexive pull-back of \(\sigma \) . More precisely, we classify all singular symplectic surfaces distinguish them in primitive symplectic surfaces, irreducible symplectic surfaces and 2-dimensional irreducible symplectic orbifolds. Moreover, we prove that the Hilbert scheme of two points on such a surface X is an irreducible symplectic variety, at least in the case where the smooth locus of X is simply connected.
