We determine equilibrium configurations of interacting point vortices in which identical vortices are arranged on concentric circles and/or form a set of nested regular polygons with or without a vortex at the center. A new analytical method is developed that uses moments of the vortex positions and yields particularly simple results for equilibria of this kind. A complete determination of all triple-ring equilibria is given and numerous previously unknown configurations are identified. Several equilibria previously reported in the literature, found by numerical solution, are characterized analytically.