Our understanding of the behaviour of electrons in glasses, liquids and non-crystalline materials generally is decades behind the detailed theory available for crystals such as silicon and germanium. This article describes the advances made in the past fifteen years. It includes the work of the Leningrad school on amorphous semiconductors, the reason why they cannot usually be doped, the concept of “variable-range” hopping and the Cohen–Fritzsche–Ovshinsky and other models for the conduction band, and the reason why glass can be transparent. A crucial experiment, that of two-dimensional conduction in an inversion layer at a Si/SiO2 interface, is described. Particular emphasis is given to concepts about which there is a disagreement and indeed controversy, such as the “mobility edge”, the “minimum metallic conductivity” and the interpretation to be given to the ovonic threshold switch.