The basic ideas and a number of results of the theory of low-energy excitations and localized electronic states, which are responsible for the universal low-temperature anomalies in glasses and unusual electronic properties of nonmetallic glasses, primarily glassy semiconductors, are reviewed. The basic hypotheses and models, previously proposed in this connection in order to explain the numerous empirical data, are discussed. Recently developed concepts regarding soft atomic configurations as a significant feature of the structure of glasses, determining new types of low-energy excitations in such anharmonic systems and of localized electronic states--self-trapped electron pairs with negative correlation energy, with very strong interelectronic attraction--are examined. The above-mentioned properties of glasses are analyzed on the basis of this theory and briefly compared with experiment. Some important problems in the theory of glasses are indicated.