  A detailed discussion of semiclassical trace formulae is presented and it is
demonstrated how a regularized trace formula can be derived while dealing only
with finite and convergent expressions. Furthermore, several applications of
trace formula techniques to quantum chaos are reviewed. Then local spectral
statistics, measuring correlations among finitely many eigenvalues, are
reviewed and a detailed semiclassical analysis of the number variance is given.
Thereafter the transition to global spectral statistics, taking correlations
among infinitely many quantum energies into account, is discussed. It is
emphasized that the resulting limit distributions depend on the way one passes
to the global scale. A conjecture on the distribution of the fluctuations of
the spectral staircase is explained in this general context and evidence
supporting the conjecture is discussed.
