Order structure of P-point ultrafilters and their relatives

We survey some recent results about the order structure of various kinds of ultrafilters. More precisely, we study Rudin-Keisler and Tukey reducibility in classes of selective, stable ordered-union, and P-point ultrafilters. Although these reductions are fundamentally different, there are connections between them. On the other hand, even though the classes of ultrafilters we consider are similar, there are significant differences in their order structure, as will be seen in the survey.