An introduction to analytical and computational methods common to problems in financial mathematics and mathematical economics. Aimed at Ph.D. students and advanced masters students who have studied stochastic calculus, the course focuses on applications of partial differential equations (PDEs) in models used in finance and economics. These include pricing financial derivatives, stochastic control problems and stochastic differential games, as well as mean field games. We discuss analytical, asymptotic, and numerical techniques for their solution.