Problems in Geotechnical Engineering are usually very complex due to the high non-linearity of the soil behaviour, the interaction with different structures, and the large deformations of the soil mass. Therefore, simulating these problems is a challenge, partly because common analysis tools employed nowadays, such as the Finite Element Method (FEM), have limitations to simulate large deformation problems. To overcome those limitations, new numerical methods have been developed in recent years, such as the Material Point Method (MPM), which has proved to be suitable for the simulation of complex geotechnical problems. The MPM combines the advantages of Eulerian and Lagrangian movement descriptions, in order to simulate large deformations problems, without the disadvantages of mesh distortion, or the presence of convective terms. This thesis presents the implementation, validation and application of an open-source computer code written by the author, based on the MPM, which may be the base of a line of research at the Universidad Nacional de Colombia. This computer code, named MPM-UN, is based on a dynamic formulation that allows to solve problems under quasi-static loads, as well as to study the dynamic nature of some geotechnical failure processes of dry and saturated soils under drained and undrained conditions, without considering changes of the pore pressures. The code integrates a frictional contact algorithm to take into account the interaction between bodies and an elastoplastic model with Mohr-Coulomb failure criteria. The validation of the code was made by simulating problems that have theoretical solutions, such as the axial vibration of an elastic bar, the bearing capacity of a continuous foundation and the sliding of a disk on an inclined plane; and by means of more complex problems such as the failure of a slope by its own weight and the collapes of a granular column. Finally, two simple slopes were simulated in order to examine the potentialities of the MPM in landslide analysis, proving that it is possible to capture complex failure behaviors with this tool, and also it allows simulating the entire deformation process, from the formation of the failure to the deposition of the material. From those simulations, the capabilities of the method to predict the generation of retrogressive failures were verified and the influence of different parameters on the run-out of the landslides was analyzed. It was found that variables that commonly are not considered, such as the angle of dilatancy and the compressibility have an important incidence in the results.