Abstract: Random events obey the principle of probability conservation, which is a fundamental and universal law of nature, paralleling the principles of energy conservation and mass conservation. Stochastic mechanics involves the aspects of stochastic structural analysis, random vibration, reliability assessment, reliability-based design optimization, and stochastic optimal control, in which the principle of probability conservation plays a fundamental and decisive role. A pair of probability density integral equations for stochastic static and dynamic systems are the unified and fundamental equations that represent the propagation and evolution of randomness. This paper describes the historical origin and current development of the principle of probability conservation, tracing its origin in statistical physics and its development in probability theory and stochastic mechanics. Then, the probability density integral equation in the time-space domain based on the principle of probability conservation is derived and established. Moreover, the theory of stochastic integral, basic idea of direct probability integral method and its recent developments in stochastic response analysis of static and dynamic systems, reliability evaluation and reliability-based optimum design are introduced, aiming to promote the advancement of research and teaching in stochastic mechanics.