Covers a large variety of subjects, from the one real variable case (with all its mathematical gems) to some of the most advanced topics such as Choquet's theory, the Prékopa-Leindler type inequalities and their ramifications, as well as the variational approach of partial differential equations and convex programming. Gentle introduction to contemporary convex function theory. Emphasized inequalities, and includes many exercises; Written at the advanced undergraduate/beginning graduate level.