Impacts involving granular masses are very common in the engineering practice. They include natural events such as landslides, as well as human-induced activities such dynamic compaction for soil improvement. Despite the different applications, these processes can be schematized in two different impact configurations: (i) impacts of blocks on still granular strata and (ii) dynamic interactions of flowing granular masses with rigid obstacles. In recent years, robust numerical codes based on continuum mechanics (Material Point Method, Particle Finite Element Method, Smoothed Particle Hydrodynamics, ecc.), able to deal with large displacements and dynamic problems, have allowed the numerical simulation of impacts, opening new frontiers on the study of these processes. In this work, MPM numerical simulations of impacts on/of granular masses are presented to discuss the necessity of a suitable constitutive relationship for obtaining reliable predictions. In particular, the constitutive model must account for (i) regime transitions between fluid-like and solid-like conditions, (ii) particle damage occurring under large stresses, and (iii) phase interactions in the case of saturated media.