We analyse the spectral correctness of Discontinuous Galerkin (DG) methods on two different examples: the Laplace eigenproblem, as a prototype of an operator with a compact inverse, and the Maxwell eigenproblem, as a prototype of an operator with a non-compact inverse. We show that, for the first example, a huge class of DG methods enjoys optimal spectral properties, whereas, for the second example, some conditions should be imposed on the DG method and on the conformity of the mesh in order to ensure spectral correctness. Numerical tests showing the sharpeness of our theory are presented: examples of optimal spectral approximation are discussed and we also comment upon DG methods and meshes which generate spourious eigenvalues.