Abstract: [Objective] Dam and dike failures occur frequently worldwide, and embankment safety is crucial for flood prevention and disaster mitigation. Research on breach evolution mechanisms and mathematical models is of great significance for flood forecasting and risk assessment. [Analysis] Following the developmental context of breach modeling, this paper summarizes the characteristics and processes of dam breaches, analyzes breach types and influencing factors, and reviews the relationship between model test results and mathematical simulations. Theoretical models, parameter models, and one-dimensional mathematical models are systematically reviewed to summarize the historical evolution and current state of dam breach modeling. The distinct features and applicability of different modeling approaches are comparatively analyzed, and common breach calculation methods are tabulated to facilitate model reference and comparison, thereby providing a clearer understanding of research directions in breach modeling. [Conclusion] In general, theoretical and parameter models are computationally simple and can be quickly applied to emergency breach assessment and disaster response, yet they fail to capture the dynamic evolution of the breach process. One-dimensional mathematical models, by incorporating complex hydraulic flow, breach geometry variation, and sediment transport, provide more detailed representations of breach dynamics, though they still rely on simplifications and assumptions of physical processes. With the rapid advancement of two- and three-dimensional mathematical models, as well as the introduction of machine learning and artificial intelligence techniques, breach modeling is evolving toward more refined physical representation and higher computational efficiency.