An outline is given of some of the steps which in the last ten years have led to an understanding of many of the properties of conduction and valence bands in non-crystalline materials. The concept of a mobility edge is described and some of the experimental evidence for its existence. A discussion is given of the nature of states in the gap, thought of as deep donors or acceptors. These pin the Fermi energy and act as recombination centres. A distinction is made between materials such as silicon or germanium with fourfold co-ordination and those like As2Te3 or SiO 2, where for the chalcogen (Te or O) co-ordination is twofold; for the former an electron in a deep state produces only moderate distortion of the lattice, for the latter the distortion is large. Photoluminescence and other properties of chalcogenides ave discussed in terms of a recent model of charged dangling bonds. It is suggested that free holes in As2Te3 and As 2Se3 are not self-trapped but that in SiO2 they are. Finally the nature of the optical absorption edge in SiO2 is contrasted with that in a soda glass in terms of a model of self-trapped excitons.