Discontinuous Galerkin methods have received considerable attention in recent years for applications to many problems in which convection and diffusion terms are present. Several alternatives for treating the diffusion flux effects have been introduced, as well as, for treatment of the convective flux terms. This report summarizes some of the treatments that have been proposed. It also considers how elementary finite volume methods may be considered as the most primative form of a discontinuous Galerkin method as well as how it may be formed as a finite element method. Several numerical examples are included in the report which summarize results for discontinuous Galerkin solutions of one-dimensional problems with a scalar variable. Results are presented for diffusion-reaction problems, convection-diffusion problems, and a special problem with a turning point. We identify aspects which relate to accuracy as well as stability of the method.