Discretization of partial differential equations (PDEs) is based on the theory of function approximation, with several key choices to be made: an integral equation formulation, or approximate solution operator; the type of discretization, defined by the function subspace in which the solution is approximated; the choice of grids, e.g. regular versus irregular grids to conform to the geometry, or static versus solution adaptive grids. We explore some of the common approaches to the choice of form of the PDE and the space-time discretization, leaving discussion of the grids for a later chapter. The goal is to introduce the reader to various forms of discretization and to illustrate the numerical performance of different methods. In particular, we will address how to choose a method that is accurate, robust and efficient for the problem at hand.