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.