We reframe much of the study of Borel reductions between orbit equivalence relations as the study of the classification strength of Polish groups. This is a partial order where we say G is stronger than H iff every orbit equivalence relation induced by a continuous action of H on a Polish space is Borel reducible to such an orbit equivalence relation induced by G. We discuss recent results pertaining to the non-Archimedean Polish groups with maximum classification strength, namely, the groups which involve S_infty. We give several surprising conditions which are equivalent to involving S_infty. Time permitting, we will discuss other work on other parts of the hierarchy of classification strength, including joint work with Aristotelis Panagiotopoulos.