The mathematical concept of supermodularity formalizes the idea of complementarity and opens the way for a rigorous treatment of monotone comparative statics and games with strategic complementarities. The approach is based on lattice methods and provides conditions under which optimal solutions to optimization problems change in a monotone way with a parameter. The theory of supermodular games exploits order properties to ensure that the best response of a player to the actions of rivals is increasing in their level. It yields strong results that apply to a wide range of games including dynamic games and games of incomplete information.