This chapter presents some of the most common numerical features used in Material Point Method simulations. The application of boundary conditions along moving boundaries in large deformation Material Point Method simulations requires special attention, especially for non-zero tractions. Two kinds of boundary conditions can be distinguished – essential and natural. Essential boundary conditions are imposed directly on the solution, and the degrees of freedom are directly eliminated from the system of equations. Natural boundary conditions are imposed on a secondary variable, such as stresses or pressures. In order to illustrate the effect of introducing an absorbing boundary, the problem of wave propagation in a linear elastic saturated porous medium is considered. To accelerate the convergence to quasi-static equilibrium, an artificial local damping can be included in the formulation in order to introduce energy dissipation in the dynamic momentum conservation. Artificial bulk viscosity damping has the purpose to improve the simulation of shock propagation.