This chapter can be conceived as a substantial course on convex analysis. But it appears here in view of its relationships with other subjects such as optimization and differential calculus. Convex functions have remarkable continuity and differentiability properties. They offer a substitute to the derivative, the subdifferential, whose calculus rules are delineated. Moreover, convexity allows rich duality properties that are displayed along two classical lines: the Lagrangian one and the perturbational one.