A central observation in quantum chaos is that
many properties of quantum systems whose classical limit is chaotic can
be modelled by random matrix theory. For example, statistical
distributions of quantum energy levels are found to agree with those
of eigenvalues of random matrices. This has been
found in an overwhelming number of numerical investigations.
In my talk I will speak about semiclassical methods
that reduce the problem of proving the connection
between quantum chaos and random matrix theory
to a problem in classical mechanics: the evaluation
of correlations between periodic orbits.