We will provide a brief survey of mathematical finance, emphasizing the relationship between stochastic processes and PDEs and their application to option pricing. Our survey will begin with Brownian motion and the Bachelier stock price model (1900), the Black-Scholes model (1973); Merton's introduction of stochastic calculus and PDE methods for option pricing (1973); Gyongy's theorem in probability (1986) and Dupire's local volatility model (1996), Heston's stochastic volatility model and degenerate parabolic PDEs (1993), and Bates stochastic volatility model with jumps (1996). We will conclude with a discussion of current research and selected open problems.