Abstract: Stochastic dynamical system analysis is a highly important research area across various science and engineering fields. The probability density evolution theory provides a fundamental framework for adressing key challenges in this domain. This paper reviews recent advances in the theory, encompassing the content, significance and historical development of the principle of preservation of probability, the generalized density evolution equation, the dimension-reduced probability density evolution equation, and related numerical methods. Finally, it also identifies important problems for further investigation.