  The present paper is devoted to developing relations between Galois \'etale
coverings in codimension 1 and \'etale fundamental groups in codimension 1 of
algebraic varieties, aimed to studying the topology of Mori dream spaces. In
particular, the universal \'etale covering in codimension 1 of a non-degenerate
toric variety and a canonical Galois \'etale covering in codimension 1 of a
Mori dream space (MDS) are exhibited. Sufficient conditions for the latter
being either still a MDS or the universal \'etale covering in codimension 1 are
given. As an application, a canonical toric embedding of K3 universal
coverings, of Enriques surfaces which are Mori dream, is described.
