Landslides and slope instabilities represent one of the most important problems in geotechnics causing significant damages around the world every year. Understanding the mechanics of the whole process is of particular importance for risk assessment. First, it is important to determine what areas may be susceptible to landsliding. In addition, it is essential to estimate the travelled distance and the velocity of the unstable mass in order to prevent severe damage. The need to develop solution schemes capable of simulating failure initiation as well as post-failure dynamics is also required in most geotechnical analyses. For instance the design of dams, tunnels, pipes, foundations or embankments. The prediction of such catastrophic episodes presents several challenges due to the complexities of real soil behaviour. In addition, the consideration of the coupled behaviour of soil and pore fluids is essential by means coupled hydromechanical formulations. Traditional geotechnical analysis, such as Limit Equilibrium Methods (LEM), and the well-known standard lagrangian Finite Element Methods (FEM) are very useful to study the failure initiation, but they provide limited information on the post-failure behaviour. In order to overcome such difficulties, modern numerical approaches are being developed. This is the case of the Material Point Method (MPM), which offers an interesting alternative. MPM discretises the media into a set of lagrangian material points which move attached to the material carrying the soil properties. Governing equations are solved incrementally at the nodes of a computational grid that remains fixed through the calculation. This dual description of the media prevents mesh distortion problems. This Thesis focusses on studying brittle failures and slope instabilities, from static conditions to run-out. Relevant aspects for the interpretation of landslides are described: the development of progressive failure mechanism, the role played by internal shearing in compound slides, and the effect of brittleness on the onset of failure and run-out. Different slope instabilities are presented. First, the Selborne slope experiment is simulated. This case, well identified with laboratory data, has been an opportunity to perform a validation of the MPM formulation. A simplified geometry of the Vajont landslide is also analysed in a second modelling. It has shown that a kinematically admissible failure mechanism requires internal shearing of the mobilised mass controlled by the geometry of the basal sliding surface. In addition, by means of a parametric study varying peak and residual strength, run-out is found to be directly related with brittleness index. Finally, a step forward in the application of MPM to multi-phase problems in porous media has been achieved. In order to simulate the behaviour of unsaturated materials, MPM has been extended by means a coupled 3-phase 1-point MPM formulation. In this way, the interaction of three different phases (solid liquid and gas) is taken into account within each material point. This approach is validated by means the modelling of an infiltration problem. Finally, an embankment slope instability induced by heavy rain has been simulated. Two constitutive models are used in the applications: a brittle model with strain softening for saturated soils, and a Mohr-Coulomb elastoplastic model formulated in terms of net stress and suction.