Determining whether or not a given finitely generated group is permutation stable is in general a difficult problem. In this talk we discuss work of Becker, Lubotzky and Thom which gives, in the case of amenable groups, a necessary and sufficient condition for permutation stability in terms of the invariant random subgroups (IRSs) of the group. We outline some ideas from the proof of this criterion, give an introduction to IRSs in general, and show how they yield new families of examples of stable and unstable groups.