  Due to their flexibility, Fox-$H$ functions are widely studied and applied to
many research topics, such as physics, statistics, viscoelasticity, etc. The
special cases represented by the Mittag-Leffler and Wright functions prove
their great impact. In this paper, we focus on certain explicit assumptions
that allow us to use the Fox-$H$ functions as densities. We then provide a
subfamily of the latter, called Fox-$H$ densities with moments of any order,
and give their Laplace transforms as entire generalized Wright functions. The
class of r.v.'s with these densities is proved to possess a monoid structure.
We present eight subclasses of special cases of such densities (together with
their Laplace transforms) that are particularly relevant in applications. To
analyze the existence conditions of Fox-$H$ functions, we derive asymptotic
results and their analytic extension.
