Snow is a complex and fascinating material which can sustain stresses like a solid or flow like a fluid depending upon the load it is subjected to. The transition from solid to fluid state can lead to catastrophic snow slab avalanches. Modeling large deformations, compaction, fractures and contacts involved in snow and avalanche mechanics requires both suitable constitutive models and an appropriate numerical framework. Recently, the Material Point Method (MPM), a hybrid Eulerian-Lagrangian continuum approach, has shown great promise in modeling complex processes involved in snow avalanche release and flow in a unified manner. This includes compaction hardening, anticrack nucleation and propagation during slab avalanche release and flow regime transition reported in avalanche dynamics. In this Chapter, we first introduce the state of the art in snow and avalanche mechanics as well as related modeling challenges. Second, we recall finite strain elastoplasticity theory and the MPM algorithm. Third, we describe different constitutive snow models based on damage or critical state soil mechanics. Finally, we show the application of MPM to snow microstructure compression, anticrack propagation in avalanche release as well as avalanche dynamics.