In this final chapter we present the discontinuous Galerkin (dG) method. This method is based on finite element spaces that consist of discontinuous piecewise polynomials defined on a partition of the computational domain. Such methods are very flexible, for example, since they allow construction of more general methods and since they allow for simple adaptation. Discontinuous Galerkin methods were originally developed for first order problems and were later extended to second order problems. We cover both categories, and derive basic stability and error estimates. Due to the discontinuous nature of the finite element space additional terms in the weak form are necessary to enforce the proper continuity conditions between adjacent elements. We also consider how to handle these additional terms in the implementation of the method.