The basic theory of material point method (MPM), which is a new meshfree method,was briefly introduced in this paper.MPM takes advantage of both Eurlerian and Lagrangian methods. It avoids the mesh distortion and tangling issues associated with Lagrangian methods and the advection errors associated with Eulerian methods. A MPM computational code called MPM-EXPLICIT with the Von-mises material strength model and Shock equation of state was developed in FORTRAN 90, and was used to compute various impact problems. The calculated result and experimental result were compared to confirm the accuracy of the code. The results obtained by using the MPM, FEM and SPH were compared. It shows that MPM possesses many prominent features. This study indicates that the material point method is an efficient and promising method for simulating the impact problems.