The foundations of modern economic theory rest on the analysis of equilibrium. The two major strands in equilibrium theory, game theory and general equilibrium theory, study rational decision makers who interact with each other directly or through the market. In each case, a fundamental question in the development of the theory is that of the existence of equilibrium. Answering this requires the application of a fixed point theorem. In 1950 Nash provided the first such application to prove the existence of (Nash) equilibrium in a non-cooperative game. Related methods led Arrow and Debreu and McKenzie in 1954 to provide conditions for the existence of a competitive equilibrium in an economy with many consumers and producers. The aim of this lecture is to provide a brief overview of these results, illustrating the centrality of mathematical tools in the development of economic theory.