A central goal of scientists and engineers is obtaining solutions of the differential equations that govern their physical systems.This can be done numerically for large and/or complex systems using finite-difference methods, finite-element methods, or spectral methods.This chapter gives an introduction and the formal basis for these methods, with particular emphasis on finite-difference methods.Second-order partial differential equations are classified as elliptic, parabolic, or hyperbolic, and the numerical methods developed for such equations must be faithful to their mathematical properties.