The current flow between two (normal) metal electrodes, separated by a vacuum gap or an insulating layer, is considered from a general point of view as a process of electron transits through and over a potential energy barrier. The main results of a unified theory of field and thermionic emission, which takes into account the band structure of the insulator, are discussed. Experiments confirming the theory both for the cases of electron emission in vacuum and in some insulators are also considered. Certain directions of future work on experimental tests of the independent-electron model for the emission process are suggested.