  This paper is devoted to extend some Hu-Keel results on Mori dream spaces
(MDS) beyond the projective setup. Namely, $\Q$-factorial algebraic varieties
with finitely generated class group and Cox ring, here called \emph{weak} Mori
dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a
neat embedding of a (completion of a) wMDS into a complete toric variety are
studied, showing that, on the one hand, those which are complete and admitting
low Picard number are always projective, hence Mori dream spaces in the sense
of Hu-Keel. On the other hand, an example of a wMDS does not admitting any neat
embedded \emph{sharp} completion (i.e. Picard number preserving) into a
complete toric variety is given, on the contrary of what Hu and Keel exhibited
for a MDS. Moreover, termination of the Mori minimal model program (MMP) for
every divisor and a classification of rational contractions for a complete wMDS
are studied, obtaining analogous conclusions as for a MDS. Finally, we give a
characterization of a wMDS arising from a small $\Q$-factorial modification of
a projective weak $\Q$-Fano variety.
