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, saddlepoint 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.