Analysis of space- and time-dependent slowing down is carried out using diffusion, P1, and B1 approximations for hydrogen and nonhydrogenous moderators. The scattering cross section is assumed to be constant, but the cross section at source neutron energy is allowed to have a different value. The calculations are primarily concerned with the first and second time moments of the flux distribution (r, v, t). The results are used to interpret the spatial variation of slowing down times. The characteristic behavior in the case of water is shown to arise from the considerably longer mean-free-path at the source energy (MeV). Time moments are also used to show that the transport approximation in P1 and B1 calculations for hydrogen can lead to a nonphysical result. Given the time moments, a simple procedure is proposed for obtaining (r, v, t). This approximation is found to be quite accurate when tested using diffusion theory moments. Comparison of theory and experiment is discussed in terms of recent measurements and Monte Carlo studies on water. So far as agreement with the available data is concerned, time-dependent diffusion theory calculations are found to be quite adequate.