Abstract: We introduce several aspects of stability of systems described by a Hamiltonian H(t)=H+V(t), where H has discrete spectrum and V(t) is a time-periodic uniformly bounded perturbation. In the first part of the lecture we introduce the notion of the monodromy and illustrate its significance for the existence of bound and propagating states. The spectral properties of the monodromy are related to those of so called Floquet Hamiltonian K=-i\partial_t+H(t). In the second part we present a result in the case when the gaps in the spectrum of H are decreasing. In the last part of the lecture we discuss dynamical stability of these systems i.e. the long-time behavior of the energy expectation value, and mention an exciting open problem, the quantum ball.