The cavitation phenomenon can be controlled to enhance the performance of high-speed marine vessels and projectiles, as well as in material processing technologies, in medicine, etc. Controlling the cavitating flows in liquids can be achieved only by advancing the mathematical foundation of the cavitation processes. These processes are manifested in different ways, the most common ones and promising for control being bubble cavitation and supercavitation. The first exact classical solution should perhaps be credited to the well-known solution by Hermann von Helmholtz in 1868. The earliest distinguished studies of academic type on the theory of a cavitating flow with free boundaries and supercavitation were published in the book Jets, wakes and cavities followed by Theory of jets of ideal fluid. Widely used in these books was the well-developed theory of conformal mappings of functions of a complex variable, allowing one to derive a large number of exact solutions of plane problems. Another venue combining the existing exact solutions with approximated and heuristic models was explored in the work Hydrodynamics of Flows with Free Boundaries that refined the applied calculation techniques based on the principle of cavity expansion independence, theory of pulsations and stability of elongated axisymmetric cavities, etc. and in Dimensionality and similarity methods in the problems of the hydromechanics of vessels.