This book covers elementary aspects of category theory and topos theory. It assumes little mathematical background, using categorical methods throughout rather than beginning with set theoretical foundations. It gives a clear exposition of key concepts and gives complete elementary proofs of theorems, including the fundamental theorem of toposes and the sheafification theorem. It ends with topos theoretic descriptions of sets, of basic differential geometry, and of recursive analysis. This book will be essential reading for third year undergraduates and graduates studying logics and category theory as part of a course on mathematics, computer science, or philosophy.