In this survey paper we present a class of shape optimization problems where
the cost function involves the solution of a PDE of elliptic type in the
unknown domain. In particular, we consider cost functions which depend on the
spectrum of an elliptic operator and we focus on the existence of an optimal
domain. The known results are presented as well as a list of still open
problems. Related fields as optimal partition problems, evolution flows,
Cheeger-type problems, are also considered.Comment: 42 pages with 8 figure
