This book presents a study of the stability of mechanical systems, i.e. their free response when they are removed from their position of equilibrium after a temporary disturbance. After reviewing the main analytical methods of the dynamical stability of systems, it highlights the fundamental difference in nature between the phenomena of forced resonance vibration of mechanical systems subjected to an imposed excitation and instabilities that characterize their free response. It specifically develops instabilities arising from the rotor–structure coupling, instability of control systems, the self-sustained instabilities associated with the presence of internal damping and instabilities related to the fluid–structure coupling for fixed and rotating structures. For an original approach following the analysis of instability phenomena, the book provides examples of solutions obtained by passive or active methods.