The physical and mathematical problems associated with radioactive waste disposal have been outlined and discussed. Some of the more important relationships and equations have been derived and explained with a view to showing how techniques developed in conventional reactor physics problems can be applied with great effect to radionuclide transport. We stress in particular the problems associated with radionuclide transport through spatially random media such as fissured and porous rock. Three distinct modeling procedures are presented: (1) the classical advective dispersion equation and its interpretation as a stochastic differential equation, (2) a purely advective approach in which the groundwater velocity and the retardation factor are random functions, and (3) an analogy with neutron transport by regarding motion along fissures and subsequent branching as a pseudo-scattering process. We describe the mathematical methods needed to solve these stochastic problems and include perturbation theory, Novikov’s theorem and the marked Brownian particle. The relationship between the methods and the non-Fickian behavior that results are discussed and used to explain the scale-dependent experimental results for the dispersion coefficient. In general, the paper attempts to be instructive in that several results are presented which are not new, but also creative in that these results are presented in a new light. Two new models are also discussed and their advantages and shortcomings outlined.