In these two talks, which are intended to be understood by a broad audience of mathematicians, I will develop the theory of cli Polish groups as they are currently known. A Polish group is a topological group where the topology is Polish (separable and completely metrizable). A Polish group is cli iff there is a compatible complete left-invariant metric. Most notably, we will show that the cli Polish groups are arranged into a hierarchy and have a natural notion of rank. From this we can conclude that there is no universal cli Polish group. Even more, we show that groups of higher rank are inherently more complicated than groups of lower rank in the sense that they produce stronger orbit equivalence relations under Borel reducibility. These results follow from work of Deissler, Malicki, and joint work with Panagiotopoulos.