Hamiltonian dynamical systems with a small number of degrees of freedom, and having ergodic classical dynamics, exhibit universal statistical laws as for their spectral fluctuations of quantal energy levels around the (non-universal) smooth average of the spectral staircase. This finding is one of the cornerstones of the stationary problems in quantum chaos: The so-called Bohigas-Giannoni-Schmit conjecture [16] states that in such cases the level statistics of the random matrix theories (of GOE if there is an antiunitary symmetry in the system, or of GUE otherwise) apply. This conjecture has been supported by many numerical experiments, but also by the semiclassical results of Berry [7], and it has recently been announced to be proven by Andreev et al [1], but under stronger conditions than ergodicity (exponential decay of classical correlations). In cases of classical integrability we also find universality, namely the statistics is typically Poissonian, although there are exceptions, whose measure somehow is expected to be zero or, at least, very small. Finally, there is the class of systems with mixed classical dynamics (regular motions coexisting in the classical phase space with chaotic regions), which are dynamically generic systems, in which case a semiclassical theory can be set up [10, 67, 53, 54]. We present recent results on this class of Hamiltonian systems.