I will survey a few papers concerning the following question: What can one learn about G from its representation theory into S. Where S=S∞ is the Polish group of all permutations of a countable set. Hom(G,S) is a Polish space in its own right. We focus on two aspects. How a (Baire) generic representation of G looks like. And the existence of representations with special transitivity properties such as faithful primitive actions, or faithful highly transitive actions of the group.