According to modern ideas there is a sharp boundary between the localized and extended states in the electron spectra of disordered systems (amorphous semiconductors, doped crystalline semiconductors, etc.). The mathematical description of the electron states near this boundary is a very complicated problem which is not yet completely solved. The purpose of this review is to describe, in the simplest possible way, without going into the mathematical problems, the basic physical ideas that have been put forward in this field. The Anderson and Lifshitz models are considered in detail and the criteria for localization are discussed. Examples are given of systems in which it is possible to observe experimentally the transition from localized to extended states that occurs when certain parameters are varied. The concept of minimum metallic conductivity and the properties of the wave functions near the boundary of the localized states are discussed. The results of computer experiments that attempt to model a disordered system near the mobility edge are described. The difficulties that arise in attempts to give a mathematical description of such a system, and the basic ideas of such a description, are discussed in the form of a conversation with an unsophisticated partner.