"Convex functions play an important role in many branches of mathematics, as well as other areas of science and engineering. The present text aims to present a thorough introduction to contemporary convex function theory, which entails a powerful and elegant interaction between analysis and geometry. A large variety of subjects is covered, from one real variable case (with all its mathematical gems) to some of the most advanced topics such as Choquet's theory, the Prekopa-Leindler type inequalities and their ramifications, the variational approach of partial differential equations and convex programming." This book can be used for a one-semester graduate course on Convex Functions and Applications, and also as a valuable reference and source of inspiration for researchers working with convexity. The only prerequisites are a background in advanced calculus and linear algebra. Each section ends with exercises, while each chapter ends with comments covering supplementary material and historical information. The authors also include many new results based on their years of experience researching convexity theory.