Let $G$ be a finite group acting the plane. The equivariant Hilbert scheme of $r$ points parametrizes $G$-equivariant finite length subschemes whose structure sheaf carries $r$ copies of the regular representation of $G$. For $G$ contained in the special linear group, equivariant Hilbert schemes have been studied extensively in the context of geometric representation theory, the Mckay correspondence, and combinatorics of Young tableaux. We will survey some of these results and then present joint work with Gjergji Zaimi on the cohomology of equivariant Hilbert schemes for $G$ an abelian group. We prove dual periodicity and quasipolynomiality behavior for the Betti numbers as $G$ varies, generalizing a geometric duality for continued fractions in the theory of toric surfaces.