Summary: Discontinuous finite elements are known to possess interesting properties of practical nature and the non-existence of theoretical obstacles can make their use very advisable in many problems. This work is concerned with the presentation of a number of theoretical results recently developed that confirm that discontinuous finite element formulations are as attractive as continuous finite element formulations from the theoretical viewpoint, in the sense that most theoretical results that hold true for classical formulations can be slightly adjusted so as to remain valid in discontinuous Galerkin formulations.