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Bringing together techniques from various areas of mathematics, this book presents an introduction to the basic concepts, methods, and results of invariant descriptive set theory. It reviews classical and effective descriptive set theory; studies Polish groups and their actions; and covers Borel reducibility results on Borel, orbit, and general definable equivalence relations. The author also describes infinitary logic, Scott analysis, and the isomorphism relation on natural classes of countable models. The book concludes with applications to classification problems and many benchmark equivalence relations. It also contains a large number of exercises at the end of most sections.