Hydrogen spillover has emerged as a possible technique for achieving high-density hydrogen storage at near-ambient conditions in lightweight, solid-state materials. We present a brief review of our combined theoretical and experimental studies on hydrogen spillover mechanisms in solid-state materials where, for the first time, the complete mechanisms that dictate hydrogen spillover processes in transition metal oxides and nanostructured graphitic carbon-based materials have been revealed. The spillover process is broken into three primary steps: (1) dissociative chemisorption of gaseous H2 on a transition metal catalyst; (2) migration of H atoms from the catalyst to the substrate and (3) diffusion of H atoms on substrate surfaces and/or in the bulk materials. In our theoretical studies, the platinum catalyst is modeled with a small Pt cluster and the catalytic activity of the cluster is examined at full H atom saturation to account for the essentially constant, high H2 pressures used in experimental studies of hydrogen spillover. Subsequently, the energetic profiles associated with H atom migrations from the catalyst to the substrates and H atom diffusion in the substrates are mapped out by calculating the minimum energy pathways. It is observed that the spillover mechanisms for the transition metal oxides and graphitic carbon-based materials are very different. Hydrogen spillover in the transition metal oxides is moderated by massive, nascent hydrogen bonding networks in the crystalline lattice, while H atom diffusion on the nanostructured graphitic carbon materials is governed mostly by physisorption of H atoms. The effects of carbon material surface curvature on the hydrogen spillover as well as on hydrogen desorption dynamics are also discussed. The proposed hydrogen spillover mechanism in carbon-based materials is consistent with our experimental observations of the solid-state catalytic hydrogenation/dehydrogenation of coronene.