We study the existence and regularity of optimal domains for functionals depending on the
spectrum of the Dirichlet Laplacian or of more general Schrödinger operators. The domains are
subject to perimeter and volume constraints we also take into account the possible presence of
geometric obstacles. We investigate the properties of the optimal sets and of the optimal state
functions. In particular we prove that the eigenfunctions are Lipschitz continuous up to the
boundary and that the optimal sets subject to the perimeter constraint have regular free
boundary. We also consider spectral optimization problems in non-Euclidean settings and
optimization problems for potentials and measures as well as multiphase and optimal partition
problems.
