It is known that molecular hydrogen is a relatively stable, ubiquitous gas that is a minor component of the atmosphere. At the same time, in recent decades molecular hydrogen has been shown to have diverse biological effects. By the end of 2022, more than 2000 articles have been published in the field of hydrogen medicine, many of which are original studies. Despite the existence of several review articles on the biology of molecular hydrogen, many aspects of the research direction remain unsystematic. Therefore, the purpose of this review was to systematize ideas about the nature, characteristics, and mechanisms of the influence of molecular hydrogen on various types of cells, including stem cells. The historical aspects of the discovery of the biological activity of molecular hydrogen are presented. The ways of administering molecular hydrogen into the body are described. The molecular, cellular, tissue, and systemic effects of hydrogen are also reviewed. Specifically, the effect of hydrogen on various types of cells, including stem cells, is addressed. The existing literature indicates that the molecular and cellular effects of hydrogen qualify it to be a potentially effective agent in regenerative medicine.