Recent years have witnessed an ever growing interest in the interactions between hydrogen atoms and a graphene sheet. Largely motivated by the possibility of modulating the electric, optical and magnetic properties of graphene, a huge number of studies have appeared recently that added to and enlarged earlier investigations on graphite and other carbon materials. In this review we give a glimpse of the many facets of this adsorption process, as they emerged from these studies. The focus is on those issues that have been addressed in detail, under carefully controlled conditions, with an emphasis on the interplay between the adatom structures, their formation dynamics and the electric, magnetic and chemical properties of the carbon sheet.