Abstract: [Objective] Dam and dike failures, as one of the most frequent disaster events worldwide, have a profound impact on human production and daily life. Research on breach calculation methods is crucial for the assessment, prediction, and risk prevention of dam-break and dike-break floods. [Analysis] From a mathematical modeling perspective, this paper reviews and summarizes two-dimensional (2D) and three-dimensional (3D) computational models for breach development. The characteristics of common 2D and 3D mathematical models are outlined, and a comparative table of typical breach calculation methods is provided. Three representative 2D mathematical models are briefly introduced to facilitate comparative studies among researchers and to better understand the current development of dam-breach mathematical models. Furthermore, commonly used numerical computation techniques, commercial software, and open-source tools are summarized and compared. The current applications and future prospects of machine learning methods in breach modeling are also discussed, along with suggestions for future research directions and key priorities. [Conclusion] Overall, existing studies on breach development mechanisms and numerical simulations still involve many simplifications and assumptions. Both 2D and 3D numerical methods are evolving toward more refined descriptions of breach development processes, but they require significant computational resources and time. Accurate and efficient full-process simulation of breach evolution is expected to remain an active research topic. Machine learning methods have gradually been introduced into breach development prediction and analysis, and they are expected to play an increasingly important and in-depth role in future research.