This paper reviews the present state of knowledge of the abundances and physical state of interstellar atomic and molecular hydrogen. Much new data in this area have been obtained in recent rocket observations. There have also been new developments as a result of ground-based infrared and 21-cm observations, and theoretical research. Rocket observations of the Lyman-α interstellar absorption line of atomic hydrogen indicate that, in many directions in the sky, atomic hydrogen is up to a factor of 10 less abundant than previously indicated by 21-cm emission measurements. In the direction of the Orion Nebula, most of the absorbing gas appears to be concentrated in the near vicinity of the nebula and to have a temperature considerably lower than the average of 100 K obtained from 21-cm emission measurements. Molecular hydrogen appears essentially absent from the general interstellar medium, as confirmed by theoretical studies of photodissociation processes. However, ground-based infrared and 21-cm studies indicate that the hydrogen in dark dust clouds is mostly molecular.