Radiation transport computations require the numerical approximation of integro-differential equations in a high-dimensional phase space. We start off by contrasting different moment methods aimed at minimizing spurious Gibbs phenomenon oscillations. We then discuss high-order methods to solve the resulting moment systems, with a particular focus on asymptotic preserving properties, meaning that the diffusive nature of radiation transport in the optically dense regime is reproduced automatically by the numerical scheme, even when under-resolved. A particularly simple approach, based on staggered grids and exponential integrators, implemented in the free and open-source software StaRMAP, is shown in a radiation dose simulation as it arises in cancer therapy. Speaker(s): Benjamin Seibold (Temple University)