Introduction to the use of convex analysis in economic theory. Includes separating hyperplane theorems, continuity and differentiability properties of convex and concave functions, support functions, subdifferentials, Fenchel conjugates, saddle-point theorem, theorems of the alternative, polyhedra, linear programming, and duality in graphs. Introduction to discrete convex analysis and matroids. Emphasis is on the finite-dimensional case, but infinite-dimensional spaces will be discussed. Applications to core convergence, cost and production functions, mathematical finance, decision theory, incentive design, and game theory.