  We review recent studies about the resonance spectrum of quantum scattering
systems, in the semiclassical limit and assuming chaotic classical dynamics.
Stationary quantum properties are related to fractal structures in the
classical phase space. We focus attention on a particular class of problems
that are chaotic maps in the torus with holes. Among the topics considered are
the fractal Weyl law, the formation of a spectral gap and the morphology of
eigenstates. We also discuss the situation when the holes are only partially
transparent and the use of random matrices for a statistical description.
