In this chapter, the issues of global stability, bifurcations, and emergence of nontrivial limiting dynamic regimes in systems described by differential equations with discontinuous right-hand sides are considered within the framework of the theory of hidden oscillations. Such systems are important in the problems of mechanics, engineering, and control, and arise both a priori and as a result of idealization of some characteristics included in real physical systems. Determining the boundaries of global stability, scenarios of its violation, as well as identifying all arising limiting oscillations are the key challenges in the design of real systems based on mathematical modeling. While the self-excitation of oscillations can be effectively investigated numerically, the identification of hidden oscillations requires special analytical and numerical methods. The analysis of hidden oscillations is necessary to determine the exact boundaries of global stability, to estimate the gap between the necessary and sufficient conditions of global stability, and their convergence. This work presents a number of theoretical results and engineering problems in which hidden oscillations (their absence or presence and location) play an important role.