Actions of Polish groups are ubiquitous in mathematics. In certain branches of ergodic theory and functional analysis, one finds a systematic study of the group of measurepreserving transformations and the unitary group. In logic, the analysis of countable models intertwines with results concerning the actions of the infinite symmetric group. This text develops the theory of Polish group actions entirely from scratch, ultimately presenting a coherent theory of the resulting orbit equivalence classes that may allow complete classification by invariants of an indicated form. The book concludes with a criterion for an orbit equivalence relation classifiable by countable structures considered up to isomorphism. This selfcontained volume offers a complete treatment of this active area of current research and develops a difficult general theory classifying a class of mathematical objects up to some relevant notion of isomorphism or equivalence. Greg Hjorth received the Carol Karp Prize for outstanding work on turbulence and countable Borel equivalence relations from the Association of Symbolic Logic. Readership Graduate students and research mathematicians interested in set theory, and topological groups and Lie groups. Reviews "Presents a beautiful example of a theory whose creation and consequences, in one way or another, involve most fields of mathematics."  Bulletin of the LMS "This monograph is an important contribution to the ongoing study ... of definable equivalence relations on Polish spaces ... essential reading for anyone interested in the subject of definable equivalence relations, but is also important for mathematicians working in areas lying outside the usual boundaries of set theory."  Zentralblatt MATH Table of Contents  An outline
 Definitions and technicalities
 Turbulence
 Classifying homeomorphisms
 Infinite dimensional group representations
 A generalized Scott analysis
 GE groups
 The dark side
 Beyond Borel
 Looking ahead
 Ordinals
 Notation
 Bibliography
 Index
