A theory of the dielectric constant arising from electrons in localised states is developed. The theory applies to randomly distributed states, such as they might occur in disordered solids, impurities in crystalline semiconductors, polymer aggregates and other materials. The theory includes many-electron effects. These, it is shown, are under certain circumstances equivalent to the classical local field effect. A practical solution of the local field problem is given for arbitrarily shaped polarizable centres which are randomly distributed in space. The orientation, however, need not be random. The theory should be particularly useful where highly non-spherical polarizable centres are encountered. It is shown that for such cases the theory can be applied to much higher densities of polarizable centres than the Clausius-Mossotti formula. Attention is paid primarily to the d.c. dielectric constant. However, the frequency dependence is found for a simple case of interacting centres. The results indicate that the interaction lowers the frequency range of the dispersion. The relevance to the problem of d.c. conductivity is briefly discussed.