This chapter shows the applicability of the two-phase single-point material point method (MPM) to the simulation of soil penetration problems such as the cone penetration test (CPT). CPT is a widely used in situ soil testing technique, which consists in pushing a conical tip into the ground at a constant rate while measuring the resistance offered by the soil to the advancement of the instrument. CPT is mainly used to determine the sub-surface stratigraphy and to estimate the geotechnical parameters of the soil for geotechnical site characterisation. CPT measurements depend on the in situ mechanical response of the soil. The numerical simulation of penetration processes is challenging because it involves large deformations, soil-structure interaction, non-linear constitutive behaviour of soil, and the interaction between the solid skeleton and the pore fluid. Standard Lagrangian Finite Element Method (FEM) is not suitable for these problems because of element distortions, but Arbitrary Lagrangian-Eulerian methods and meshless methods such as MPM have been successfully applied.