A Polish group has the generic point property if every minimal flow has a comeager orbit. It has been shown that any Polish group with metrizable universal minimal flow has the generic point property, but the converse does not hold, as shown by Kwiatkowska. Until recently, Kwiatkowska’s counterexample was the only known instance of this phenomenon. We will discuss some properties and many questions involving the class of groups with the generic point property, and describe a new family of counterexamples, obtained in a joint work with Todor Tsankov.