Although a large number of approaches have been employed for numerical solutions of PDEs, quite straightforward FD-based approximations remain of great utility and importance. Just as for ODEs, high orders of accuracy often significantly increase computational efficiency. This chapter describes FD approaches for a variety of different types of PDEs, together with both basic theory and practical considerations. For many commonly occurring equations (e.g., Laplace, Poisson, and Helmholtz equations), their analytic character permits more effective FD approximations than what one obtains by simply approximating each derivative separately.