A new technique is developed in this thesis, which applies boundary conditions to moving boundaries in the Material Point Method (MPM). While MPM has been proven to be useful in slope stability, foundation and seabed ploughing modelling, the application of boundary conditions is still a challenge, because the location of the boundary is unknown in MPM. Boundary conditions have currently been applied to fixed boundaries with a known location. These conditions can be applied directly onto the background grid. In this thesis the concept of applying boundary conditions to the background grid has been expanded to boundaries, which do not coincide with the background grid. This technique can thus also be used for moving boundary conditions. Therefore, the research question of this thesis is: Can boundary conditions on moving boundaries be appropriately applied to the background mesh of an MPM in slope stability problems? The location of the boundary is not defined by current versions of MPM. Thus an edge detection method was constructed to locate the boundary based on the information provided by MPM. The Volume of Fluid method (VOF) and the Surface Marker Method (SMM), two edge detection methods used in fluid dynamics, have been tested. Moreover, a new implementation of the Level Set Method has been developed in this thesis, which uses a composite Bézier curve to define the boundary. The method has been called the Proximity Field Method (PFM). PFM locates the boundary based on the distance towards a material point, VOF uses the volume of material at a location to determine the location of boundaries and SMM places fake material points at the initial boundary to track it. The accuracy of VOF proved to be too small for the application of boundary conditions. SMM gave a better representation of the boundary at small strains in comparison to PFM. However, the main advantage of MPM is the possibility of large strain modelling, which SMM could not handle. PFM was able to handle these larger strains and was therefore chosen as the basis for the application of boundary conditions. PFM’s representation of the boundary was improved to be useful in the application of boundary conditions. Smoothing of the surface together with an equal spacing of the material points in the initial condition have improved the representation of the boundary. Moreover, the computation cost had to be reduced, because it was too high after the initial implementation. Finally, PFM has been used to apply a surface traction to MPM. The surface traction must be distributed from the boundary to the nodes of the background grid. This technique has been tested by applying a load to the top of a soil slope. As intended, the load is transferred to the background grid and changes according to the location of the boundary. As expected, the deformation of the slope is increased due to the application of the load. So, the concept of applying boundary conditions on moving boundaries to the background grid has been proven to work.