[en] The distribution of electron momenta (electron density in momentum representation) of gases can be probed by Compton scattering of either photons (γ-rays or X-rays) or electrons. Recently it has been shown that Compton scattering of electrons is suited to the study of the electron momentum densities of solids on a microscopic scale. This technique, known as ECOSS, Electron Compton Scattering from solids can be done in the electron microscope by Electron Energy Loss Spectroscopy (EELS). The advantages and disadvantages of ECOSS as compared to photon Compton Scattering are discussed. It is shown that ECOSS is a method suitable to obtain information on the ground state wave function in condensed matter. After discussion of inherent approximations, especially the impulse approximation, the reciprocal form factor is introduced. A method is proposed in order to cope with the main difficulty, namely multiple scattering. Important applications of ECOSS are the study of anisotropy of momentum densities; correlation effects of conduction electrons in metals; and charge transfer in alloys