Real-time safety assessment (RTSA) of dynamic systems is a critical task that has significant implications for various fields such as industrial and transportation applications, especially in non-stationary environments. However, the absence of a comprehensive review of real-time safety assessment methods in non-stationary environments impedes the progress and refinement of related methods. In this paper, a review of methods and techniques for RTSA tasks in non-stationary environments is provided. Specifically, the background and significance of RTSA approaches in non-stationary environments are firstly highlighted. We then present a problem description that covers the definition, classification, and main challenges. We review recent developments in related technologies such as online active learning, online semi-supervised learning, online transfer learning, and online anomaly detection. Finally, we discuss future outlooks and potential directions for further research. Our review aims to provide a comprehensive and up-to-date overview of real-time safety assessment methods in non-stationary environments, which can serve as a valuable resource for researchers and practitioners in this field.