In this work we study the application of the Method of Fundamental Solutions (MFS) for the numerical solution of eigenvalue problems in partial differential equations. The MFS is a meshless method that was previously applied only to the calculation of eigenfrequencies for domains with simple geometries and Dirichlet boundary conditions (cf. [Karageorghis 2001]). In this work we show that a particular choice of the point-sources allow to obtain excellent results for a fairly general class of domains. We consider Dirichlet and Neumann boundary conditions for the eigenvalue problem associated to the Laplace operator in the interior and exterior cases for two-dimensional domains and for the interior case in three-dimensional domains. The case of domains with corners and cracks is also addressed enriching the MFS base of functions with some particular solutions adapted to these domains. We also present results of the application of the MFS to the eigenvalue problem associated to the Bilaplacian operator and to the Lamé operator, in the elastic case.