In this talk we discuss spectral perturbation theory from an abstract perspective. More precisely, we consider a one-parameter family of varying bilinear forms, each of them defined on a (possibly) different Hilbert space. Assuming the stability of the corresponding spectra, in the first part of the talk we discuss a quantification of the rate of convergence. A key feature is the explicit variational characterization of the first term in the asymptotic expansion of the perturbed eigenvalues, which depends on the limit eigenspace and of the magnitude of the perturbation through the reso- lution of a minimization problem. In the second part of the talk, we explore some concrete applications of our abstract results. First, we consider eigenvalue problems for the Laplace-Beltrami operator with varying measure weights; secondly, we in- vestigate the Neumann approximation of the Steklov eigenvalues of the Laplacian; finally, we focus on how the spectrum of the Laplace-Beltrami operator on a Rie- mannian manifold changes when a second small manifold is glued on a small portion of it. This is a joint project with Andrea Bisterzo, Roberto Ognibene and Prasun Roychowdhury.