  We study the moduli space of a product of stable varieties over the field of
complex numbers, as defined via the minimal model program. Our main results
are: (a) taking products gives a well-defined morphism from the product of
moduli spaces of stable varieties to the moduli space of a product of stable
varieties, (b) this map is always finite \'etale, and (c) this map very often
is an isomorphism. Our results generalize and complete the work of Van Opstall
in dimension 1. The local results rely on a study of the cotangent complex
using some derived algebro-geometric methods, while the global ones use some
differential-geometric input.
