Topos theory has led to unexpected connections between classical and constructive mathematics. This text explores Lawvere and Tierney s concept of topos theory a development in category theory that unites important but seemingly diverse notions from algebraic geometry set theory and intuitionistic logic. A virtually self-contained introduction this volume presents toposes as the models of theories -- known as local set theories -- formulated within a typed intuitionistic logic. The introductory chapter explores elements of category theory including limits and colimits functors adjunctions Cartesian closed categories and Galois connections. Succeeding chapters examine the concept of topos local set theories fundamental properties of toposes sheaves locale-valued sets and natural and real numbers in local set theories. An epilogue surveys the wider significance of topos theory and the text concludes with helpful supplements including an appendix historical and bibliographical notes references and indexes.