Hydrogen and deuterium atoms have been studied by pulsed EPR spectroscopy in polycrystalline samples of H2O ice, D2O ice, and various isotopic mixtures. At high temperature ( - 10-degrees-C) the pattern and the time-dependence of the EPR line intensities are similar to previous results for H and D in liquid water. Chemically induced dynamic electron polarization (CIDEP) is generated in second-order atom combination reactions. The CIDEP behavior was found to change over the temperature range studied ( - 5-degrees-C to 130-degrees-C), consistent with additional contributions from spur reactions, and at lower temperatures, geminate recombination of (D...OD) radical pairs. Transverse spin relaxation times were measured by the spin-echo technique, and interpreted in terms of translational motion of free atoms diffusing through the ice lattice. One surprising result is that D atoms diffuse faster than H atoms below 200 K. This is explained as a vibrational zero point energy effect, by applying transition state theory to a model in which the diffusing atom must pass through a tight "bottleneck" in the electronic potential surface, as it passes from one minimum energy site in the lattice to the next. The H and D spin relaxation rates were successfully simulated by means of a semi-classical potential which was constructed by pairwise addition of atom-atom contributions represented by modified Buckingham potential functions. Extension of the model to include tunneling resulted in little. change to the fit of the H and D data. Although predictions of muonium diffusion rates using the same potential do not give quantitative agreement with published results from spin relaxation measurements, they do serve to illustrate the dominant effect of tunneling over a wide temperature range for that light atom.