The paper is devoted to the mathematical modeling of the dynamics of geophysical flows on mountain slopes, e.g., rapid landslides, debris flows, avalanches, lava flows, etc. Such flows can be very dangerous for people and various objects. A brief description is given of models that have been used so far, as well as of new, more sophisticated, models, including those developed by the authors. In these new models, nonlinear rheological properties of the moving medium, entrainment of the underlying material, and the turbulence are taken into account. The results of test simulations of flows down long homogeneous slopes are presented, which demonstrate the influence of rheological properties, as well as of turbulence and mass entrainment, on the behavior of the flow.
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