We discuss the numerical approximation of eigenvalue problems arising from partial differential equations. In the case of Galerkin approximation of elliptic partial differential equations, the conditions for the convergence of eigenvalues and eigenfunctions are the same as for the convergence of the solutions to the corresponding source problem. We review the basic argument of the analysis which is often referred to as Babuska-Osborn theory. Then, we consider less standard schemes, including non-conforming approximations and the finite element discretization of eigenvalue problems in mixed forms. In this last example, in particular, it will be shown that standard conditions for the source problem (like the well-known Babuska-Brezzi inf-sup condition) are not sufficient for the good convergence of eigenvalues and eigenfunctions. Finally, we shall discuss the numerical approximation of eigenvalue problems in the setting of differential forms. This includes the study of the Hodge-Laplace eigenvalue problem in the framework of de Rham complex.