A review is given of the physical properties and of the theoretical models for transition metal compounds of the A 15 (or $\beta$-tungsten) structure. Not only binary compounds of stoichiometeric composition are considered, but also their multiple alloys with one another. The behavior of the electronic specific heat, magnetic susceptibility, Knight shift, and other electronic properties as functions of the composition and temperature is discussed, and a comparison is made of these results with the optical and x-ray spectra. The anomalies of the elastic constants in V$_3$Si and Nb$_3$Sn are investigated in detail, and the nature of the martensitic transformation in them is analyzed. The superconducting transition temperature as a function of the composition, stoichiometry, electron concentration, and pressure is analyzed in considerable detail for both pure compounds and for their alloys. An account is given of the existing theoretical models for the electronic structure of these compounds, and the predictions of these models with regard to electronic and lattice properties are compared with the experimental data. It appears that none of the existing models is able to explain the entire set of observed phenomena. The direction of future experimental investigations is indicated.