Progress in the study of a new class of chemically bound compounds of noble-gas atoms is reviewed. The focus is on rare-gas molecules of the form HNgY, where Ng is a noble-gas atom and Y is an electronegative group, prepared by photolysis of HY in the rare-gas matrix. Other related types of new molecules of noble-gas atoms are discussed as well. Topics discussed in this review include: (a) The nature of bonding and the energetic stability of the compounds. (b) The vibrational spectroscopy of the molecules, and its role in identification of the species. (c) The mechanism and dynamics of photochemical formation of HNgY in the matrix, and the pathways for thermal and infrared (IR)-induced decomposition. Specifically, attention is given to the issue of "direct" formation following photolysis of HY versus "delayed" formation involving H atom diffusion. (d) Molecules of the lighter rare gases Ar, Ne, and He, focusing on the experimentally prepared HArF and on theoretical predictions suggesting the existence of other molecules. (e) The most-recently discovered photochemically induced insertion compounds of Ng into hydrocarbons, such as HXeCCH. (f) Clusters of HNgY with other molecules. The possible existence of neat aggregates and crystals of HNgY. The reviewed state-of-the-art suggests this field is at an early stage of development with major open questions bearing on the surprising properties of the molecules and on the formation mechanisms. These are part of the challenge for the future.