  In the present survey paper, we present several new classes of Hochster's
spectral spaces "occurring in nature", actually in multiplicative ideal theory,
and not linked to or realized in an explicit way by prime spectra of rings. The
general setting is the space of the semistar operations (of finite type),
endowed with a Zariski-like topology, which turns out to be a natural
topological extension of the space of the overrings of an integral domain,
endowed with a topology introduced by Zariski. One of the key tool is a recent
characterization of spectral spaces, based on the ultrafilter topology, given
in a paper by C. Finocchiaro in Comm. Algebra 2014. Several applications are
also discussed.
