  In this paper we classify all singular 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-\'etale covering has the algebra of reflexive forms spanned by the
reflexive pull-back of $\sigma$. We moreover 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.
