Coherent and systematic, this text explores the behavior of linear and nonlinear dynamical systems subject to parametric random vibrations, by means of the theory of stochastic processes, stochastic differential equations, and applied dynamics. It distills decades of research to formulate new stochastic stability theorems and analytical techniques for determining the random response of nonlinear systems. In addition, it resolves controversies and paradoxes related to the interpretation of certain stochastic processes and the use of analytical methods.