  We study uniformly recurrent subgroups (URS) introduced by Glasner and Weiss
\cite{GW}. Answering their query we show that any URS $Z$ of a finitely
generated group is the stability system of a minimal $Z$-proper action. We also
show that for any sofic $URS$ $Z$ there is a $Z$-proper action admitting an
invariant measure. We prove that for a $URS$ $Z$ all $Z$-proper actions admits
an invariant measure if and only if $Z$ is coamenable. In the second part of
the paper we study the separable $\C^*$-algebras associated to URS's. We prove
that if an URS is generic then its $\C^*$-algebra is simple. We give various
examples of generic URS's with exact and nuclear $\C^*$-algebras and an example
of a URS $Z$ for which the associated simple $\C^*$-algebra is not exact and
not even locally reflexive, in particular, it admits both a uniformly amenable
trace and a nonuniformly amenable trace.
