The theory of inconsistency has been growing steadily over the last two decades. One focus has been philosophical issues arising from the paradoxes of set theory and semantics. A second focus has been the study of paraconsistent or inconsistency-tolerant logics. A third focus has been the application of paraconsistent logics to problems in artificial intelligence. This book focuses on a fourth aspect: the construction of mathematical theories in which contradictions occur, and the investigation of their properties. The inconsistent approach provides a distinctive perspective on the various number systems, order differential and integral calculus, discontinuous changes, inconsistent systems of linear equations, projective geometry, topology and category theory. The final chapter outlines several known results concerning paradoxes in the foundations of set theory and semantics. The book begins with an informal chapter which summarises the main results nontechnically, and draws philosophical implications from them. This volume will be of interest to advanced undergraduates, graduate students and professionals in the areas of logic, philosophy, mathematics and theoretical computer science.