Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a better understanding of their roots.This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers the basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications.A reader should have some elementary knowledge of these three subjects, or at least two of them, in order to be able to follow the main examples, appreciate the unifying power of the categorical approach, and discover the subterranean links brought to light and formalised by this perspective.Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications in Algebra and Topology, with a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields.In this second edition, the book has been entirely reviewed, adding many applications and exercises. All non-obvious exercises have now a solution (or a reference, in the case of an advanced topic); solutions are now collected in the last chapter.Key Features: o The importance of Category Theory in Mathematics, in theoretical Computer Science and in parts of theoretical Physics is growingo The main notions of Category Theory are here presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and Topologyo Then the theory is developed presenting other examples and some 400 exercises; the latter are endowed with a solution, or a referenceo Three chapters and some extra sections are devoted to applicationso Many examples and applications are standard, others may come out as unexpected and intriguing, suggesting further investigation; a beginner should get, from these examples, exercises and applications, a concrete grasp of a theory which might risk of being quite abstracto The new edition has been rewritten in a clearer, more explicit way, adding many applications and exercises

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