electric and magnetic fields, barium in an electric field, and helium in an electric field. One-dimensional systems and potential For the case of one-dimensional system with the boundary condition the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by here is the density of states and V(x) is the classical potential of the particle, the half derivative of the inverse of the potential is related to the density of states as in the Wu-Sprung potential. Recent directions One open question remains understanding quantum chaos in systems that have finite-dimensional local Hilbert spaces for which standard semiclassical limits do not apply. Recent works allowed for studying analytically such quantum many-body systems. The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions of various chaotic Hamiltonian. For example, before the existence of scars was reported, eigenstates of a classically chaotic system were conjectured to fill the available phase space evenly, up to random fluctuations and energy conservation (Quantum ergodicity). However, a quantum eigenstate of a classically chaotic system can be scarred: the probability density of the eigenstate is enhanced in the neighborhood of a periodic orbit, above the classical, statistically expected density along the orbit (scars). In particular, scars are both a striking visual example of classical-quantum correspondence away from the usual classical limit, and a useful example of a quantum suppression of chaos. For example, this is evident in the perturbation-induced quantum scarring: More specifically, in quantum dots perturbed by local potential bumps (impurities), some of the eigenstates are strongly scarred along periodic orbits of unperturbed classical counterpart. Further studies concern the parametric () dependence of the Hamiltonian, as reflected in e.g. the statistics of avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: the standard map and the kicked rotator are considered to be prototype problems. Works are also focused in the study of driven chaotic systems, where the Hamiltonian is time dependent, in particular in the adiabatic and in the linear response regimes. There is also significant effort focused on formulating ideas of quantum chaos for strongly-interacting many-body quantum systems far from