The development of the theory of absolute stability, the theory of bifurcations, the theory of chaos, theory of robust control, and new computing technologies has made it possible to take a fresh look at a number of well-known theoretical and practical problems in the analysis of multidimensional control systems, which led to the emergence of the theory of hidden oscillations, which represents the genesis of the modern era of Andronov’s theory of oscillations. The theory of hidden oscillations is based on a new classification of oscillations as self-excited or hidden. While the self-excitation of oscillations can be effectively investigated analytically and numerically, revealing a hidden oscillation requires the development of special analytical and numerical methods and also it is necessary to determine the exact boundaries of global stability, to analyze and reduce the gap between the necessary and sufficient conditions for global stability, and distinguish classes of control systems for which these conditions coincide. This survey discusses well-known theoretical and engineering problems in which hidden oscillations (their absence or presence and location) play an important role.