Moraine dams formed by the accumulation of loose glacial materials, typically exhibit poor consolidation properties and complex structure characteristics (especially with buried ice) which make them susceptible to failure. Influenced by global climate warming, the risk of moraine dam breaching is further exacerbated. This article summarizes the breaching modes, breaching mechanisms, and breaching numerical models of moraine dams, and analyzes several deficiencies: 1) The accuracy of long-term monitoring of moraine dams is relatively low, the dynamic monitoring for individual dam and its environment is insufficient; 2) Systematic research on how buried ice affects dam breaching is lacking, and a significant gap remains in the study of combined multiple breaching modes of moraine dams; 3) The predictive results of moraine dam breaching parameter models are quite unreliable, and physically-based mathematical models have not consider the influence of buried ice. Based on this, the article puts forward the following recommendations: 1) Enhance the understanding of climate change impacts on moraine dam breaching through long-term dynamic monitoring of moraine dams; 2) Focus on the erosion characteristics of materials, study the breaching process and mechanisms of ice-rich moraine dams under various breaching modes; 3) Reveal the response mechanism of moraine dams to temperature variations using numerical simulation techniques that couple thermal-stress modules and consider the phase change of ice, further revealing the breaching mechanisms of moraine dams containing buried ice.