A Polish (i.e. separable and completely metrizable) topological group G has ample generics if every diagonal conjugation action of G on its finite power has a comeager orbit. In recent years, this notion has drawn attention of many researchers, mainly because of its very interesting and strong consequences. For example, a Polish group with ample generics has the automatic continuity property (i.e. all its homomorphisms into separable groups are continuous), it admits a unique Polish groups topology, and it has the small index property (i.e. all its subgroups with index less than continuum are open.)

In this talk, I will give an overview of results on Polish groups with ample generics, including natural examples of such groups, and discuss some future directions of research.