As financial contracts become increasingly complex, the need for robust, efficient and flexible pricing methods becomes critical for effective risk management. For many problems of practical interest, analytic solutions are infeasible. Effective numerical pricing methods are therefore required. The primary focus of this presentation will be on pricing options via partial differential equations (PDEs). We begin by comparing the advantages and disadvantages of the PDE, Monte Carlo, and lattice approaches. Details of the PDE method will then be provided. In particular, a variety of time and spatial discretization schemes will be presented for both one and two factor option pricing problems. Theoretical properties, and practical demonstrations of each scheme, will be shown. A variety of numerical issues which arise for financial contracts are also addressed, such as using penalty methods to price American options, and the effect of non-smooth payoff data on the PDE solution. Finally, methods for extending the basic PDE approach to path-dependent options and non-linear contracts will be discussed. Throughout the presentation, numerical examples will be provided to reinforce the underlying concepts.