There has been a significant growth in applications of the material point method (MPM) and its improved versions. As the methods are being applied to increasingly broad and difficult problems, new challenges to the methods arise. Some of them are related to the methods themselves, and others are related to pre- or post- processors for the methods. In this talk, we will discuss selected issues recently encountered in our applications. They are the generation of material points and presentation of the results for cases involving complex geometries, mesh and orientation dependency of the results, inconsistence of the methods when applied to problems with large stress, and properties of the solution near an interface of materials. We will discuss satisfactory solutions to some of the issues and unresolved difficulties for others. For the pre- and post- processing of the numerical results, we benchmark the significant efficiency improvements of a new material point generation method and then discuss the presentation of numerical results using STL file format on material surfaces. We will also show that MPM and its improved versions, including the dual domain material point (DDMP) method, are inconsistent for cases with large stresses and then present a solution to the issue. The different properties of MPM and DDMP results under mesh rotation will be discussed. The numerical noise of the MPM and DDMP results near a material interface for thermal and mechanical problems will be studied, and a method to address this problem is proposed. This work is sponsored by ASC, C2, ECP, and LDRD programs of Los Alamos National Laboratory under auspices of U.S. Department of Energy.