Hydrogen and its interaction with metal oxide surfaces is of major importance for a wide range of research and applied fields spanning from catalysis, energy storage, microelectronics, to metallurgy. This paper reviews state of the art of first principles calculations on the well-known ruthenium oxide (RuO2) surface in its (110) orientation and its interaction with hydrogen. In addition to it, the paper also fills gaps in knowledge with new calculations and results on the (001) surface. Bulk and surface interactions are thoroughly reviewed. This includes systematic analysis of adsorption sites, local agglomeration propensity of hydrogen, and migration pathways in which literature data and their potential deviations are explained. We notably discuss novel results on propensity for agglomeration of hydrogen within bulk channels [001] oriented in which the proton-like behavior of adsorbed hydrogen hinders further agglomeration in adjacent channels. The paper brings new insights into the migration pathways on the surface and in bulk, both exhibiting preferential diffusion paths along the [001] direction. The paper finally investigates the subsurface region. We show that while the subsurface has more stable sites for adsorption compared to bulk, its accessibility from the surface shows prohibitive activation barriers inhibiting penetration into subsurface and bulk. We further calculate and discuss adsorption and penetration processes on the alternative RuO2 (001) surface.