  In this paper we establish new quantitative stability estimates with respect
to domain perturbations for all the eigenvalues of both the Neumann and the
Dirichlet Laplacian. Our main results follow from an abstract lemma stating
that it is actually sufficient to provide an estimate on suitable projection
operators. Whereas this lemma could be applied under different regularity
assumptions on the domain, here we use it to estimate the spectrum in Lipschitz
and in so-called Reifenberg-flat domains. Our argument also relies on suitable
extension techniques and on an estimate on the decay of the eigenfunctions at
the boundary which could be interpreted as a boundary regularity result.
