Introduction.- Part I: Analysis of the discontinuous Galerkin method.-DGM for elliptic problems.- Methods based on a mixed formulation.- DGM for convection-diffusion problems.- Space-time discretization by multi-step methods.- Space-time discontinuous Galerkin method .- Generalization of the DGM.- Part II: Applications of the discontinuous Galerkin method.- Inviscid compressible flow.- Viscous compressible flow.- Fluid-structure interaction.- References.- Index.