This text provides a comprehensive introduction to convex sets and functions. Requiring only a basic knowledge of analysis and linear algebra, the book clearly discusses topics as diverse as number theory, classical extremum problems, combinatorial geometry, linear programming, game theory, polytopes, bodies of constant width, the gamma function, minimax approximation, and the theory of linear, classical, and matrix inequalities. The book clearly shows how convexity links many different topics in mathematics, from linear algebra to analysis. Suitable for upper-level undergraduate and graduate students, this book offers full solutions to over 200 exercises as well as detailed suggestions for further reading.