Experimental measurements of bulk diffusion in ice and surface diffusion on ice were performed using laser resonant desorption (LRD) techniques. Bulk diffusion in ice was examined using ice sandwich structures and continuous source experiments together with LRD depth-profiling analysis. Surface diffusion was monitored using prepare-refill-probe LRD experiments. New experimental results were obtained for the bulk diffusion of NH3 and CH3OH. These species probably exist as hydrates in the ice. The LRD measurements for CH3OH hydrate diffusion, combined with previous results, provide evidence for a vacancy-mediated diffusion mechanism. The diffusion rates for NH3 hydrates are much larger than diffusion rates for H2O self-diffusion in ice and are attributed to the disruption of the ice lattice. LRD prepare-refill-probe experiments revealed that surface diffusion was not measurable for almost all of the species examined on ice. Only butane displayed a measurable surface mobility that was attributed to its unique size and chemical nature. These new measurements of bulk diffusion in ice and surface diffusion on ice should be useful in developing our understanding of kinetic processes in and on ice that are relevant to heterogeneous atmospheric chemistry and ice core analysis.