The finite element method (FEM) is popular in cutting process simulations. Due to large strains and element distortions the FEM simulations are confronted with numerical difficulties. In the present contribution the material point method (MPM) is developed for cutting process simulations. Due to special working condition in cutting, a special contact algorithm has been applied for the simulation. Verification of computations confirm that the MPM generates realistic chip morphologies. The plastic strain and temperature distributions are compared with the FEM computations. It is found that the MPM provides a smoother chip formation and less strain localizations. The predictions on friction effect and feed influence agree with experimental observations. The investigation shows that the material point method is an efficient alternative method to the finite element methods.