Using a pedagogical, unified approach, this book presents both the analytic and combinatorial aspects of convexity and its applications in optimization. On the structural side, this is done via an exposition of classical convex analysis and geometry, along with polyhedral theory and geometry of numbers. On the algorithmic/optimization side, this is done by the first ever exposition of the theory of general mixed-integer convex optimization in a textbook setting. Classical continuous convex optimization and pure integer convex optimization are presented as special cases, without compromising on the depth of either of these areas. For this purpose, several new developments from the past decade are presented for the first time outside technical research articles: discrete Helly numbers, new insights into sublinear functions, and best known bounds on the information and algorithmic complexity of mixed-integer convex optimization. Pedagogical explanations and more than 300 exercises make this book ideal for students and researchers.