The mathematical foundations are analyzed along with the basic concepts of stability, conservative systems, pseudoconservative systems, gyroscopic systems, and circulatory systems. Special attention is paid to the extreme properties of eigenvalues, Sylvester's law of inertia and its quantitative formulation, Lyapunov's definition of stability, linear systems and stability criteria, multiparameter systems and their associated definitions, the stability of conservative systems, the convexity of the fundamental characteristic surface, the application of the Rayleigh quotient, the effect of damping, continuous systems, and other related topics.