Moduli spaces of flat Riemannian metrics on orbifolds

We study moduli spaces of flat metrics on closed Riemannian orbifolds admitting such metrics. We show that for such orbifolds O, the Teichmüller space of flat metrics Tflat(O) serves as a classifying space for proper actions of the mapping class group, MCG(O), in the appropriate category. We show that the moduli space of flat metrics, Mflat(O) = Tflat(O) / MCG(O), is itself a very good orbifold, and, under a technical assumption, describe this orbifold algebraically up to commensurability.