The subject area referred to as "wave chaos", "quantum chaos" or "quantum chaology" has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability etc. After giving a rough account on "what is quantum chaos?", I intend to list some pending questions, some of them having been raised a long time ago, some others more recent. The choice of problems (and of references) is of course partial and personal. 1 0.1. A brief overview of "quantum chaos". Let us start by resolving the apparent paradox of the phrase "quantum chaos". A classically chaotic system is defined by an ordinary differential equation, or a discrete-time map acting on some finite-dimensional phase space; for the dynamics to be chaotic, the ODE or the map is generally nonlinear in the phase space coordinates. By contrast, the associated quantum system is defined by a linear operator acting on the quantum Hilbert space; the quantum dynamics generated by this operator (either a selfadjoint Hamiltonian, or a unitary propagator) is by essence linear, but acts in a space of infinite (or at least large) dimension. Let us assume that the spectrum of that operator is discrete: the quantum dynamics is then quasiperiodic, and can be easily expressed in terms of the eigenvalues and eigenstates of the operator. One is thus led, from a time-dependent (dynamical) problem, to a time-independent (spectral) problem, which forms the "backbone" of quantum chaos:What are the spectral properties of a quantum Hamiltonian, the classical limit of which generates a chaotic dynamics?What do the corresponding eigenstates look like?These two questions are deliberately vague. Below I will give more precise formulations. To fix ideas, we will consider one of the simplest chaotic systems, namely the geodesic flow on a compact Riemannian surface (X, g) of negative curvature (and unit area). The corresponding quantum Hamiltonian is defined as the generator of the wave group on X, 1 This contribution was mostly written while I enjoyed a stay at the Centre de Recherches Mathématiques in Montréal, invited by D.Jakobson and I.Polterovich, to whom I am thankful.