In this talk we review some recent results on a posteriori error estimates for eigenvalue problem. We consider both residual type error estimators and non-residual type based on gradient reconstruction in the spirit of Prager-Synge theory. We consider the mixed formulation of Laplace equation and we extend our results, in the case of hte residual error estimator to the approximation of the eigensolutions of the Maxwell system. Moreover, we are going to present some new a priori estimates for parameter dependent eigenvalue problems.