This paper describes the development of snow avalanche dynamics models (restricted to the flow type) from the 1930s in the former Soviet Union up to 1999, when the last catastrophic avalanche winter in the European Alps occurred. The scope cannot be a comprehensive history; the paper is not much more than a selection of topics which I think are essential for avalanche hazard mapping. Emphasis is given to the different assumptions taken as a basis for models, which seems decisive for a success. The rest-analytical or numerical calculations - is routine work and not real research. Calibration of models by numerous field observations in different topography, climate and involved mass is an important aspect of application. To observe avalanches in only one avalanche path helps to some extent, but is certainly unsatisfactory. The presented models are all-up to the present day-somehow uncertain. Therefore, only relative simple models with few parameters are significant. An increase of complexity of models does not necessarily mean an increase of accuracy or a better hazard mitigation strategy.