Periodic groups saturated with finite simple groups

Let $\mathfrak{M}$ be a set of groups. We say that a group $G$ is saturated with the groups of $\mathfrak{M}$, if every finite subgroup of $G$ lies in a subgroup isomorphic to a group of $\mathfrak{M}$. We give a short survey on results about groups saturated with different sets of finite simple non-abelian groups.