This chapter outlines the most recent developments that largely enhance the mathematical accuracy of the method, while keeping the physical qualities and computational efficiency close to those of original material point method. It explains the mass lumping procedure and presents the computational steps of the widely used Modified-Update-Stress-Last (MUSL) scheme. For simplicity, only one-dimensional deformations of a one-phase continuum are considered. B-spline Material Point Method replaces the piecewise-linear basis functions by higher-order B-splines. The MPM mapping can lead to significant numerical errors, especially when large deformations are considered. For this reason, the Taylor Least Squares technique is discussed in this chapter. Direct solvers can be used though in a different setting, for example as a preconditioner. Alternatively, iterative solvers can be adopted, in which an initial guess is updated successively until a converged end-of-step solution has been reached. The new approach uses the isogeometric analysis (IgA) formulation of B-splines based on the Cox-de-Boor formula.