A Polish space (group) is a separable, completely metrizable topological space (group). This book is about actions of Polish groups, in connection with--or from the point of view of--the subject of descriptive set theory. Descriptive set theory is the study of definable sets and functions in Polish spaces. The basic classes of definable sets are the classes of Borel, analytic and coanalytic sets, and these constitute the main topic of the book, but the authors also consider other classes of definable sets. This will be a valuable book for all researchers in set theory and related areas.