The purpose of this review is to give a systematic exposition of the basic physical ideas which underlie experiments to test the law of conservation of electric charge in elementary particle interaction processes, as well as experiments to demonstrate the quantization of the electric charges of all known elementary particles. A discussion is given of experiments which demonstrate with high accuracy the equality of the absolute values of the electric charges of the proton, electron, and muon, as well as the neutrality of atoms, the neutron, the electronic and muonic neutrinos, and the photon. Results are quoted for measurements of the magnetic moments of the electron and proton, and a discussion is given of the method of using these data to determine the absolute values of the electric charge of the proton, which is one of the fundamental physical constants currently known with an accuracy of the order of several parts per million. A possible relationship between the quantization of electric charge and the existence of magnetic monopoles is examined. This review also gives the results of experimental searches for electric dipole moments of elementary particles: the neutron, proton, electron, muon, $\Lambda$ hyperon, and neutrino. The survey of the literature is complete up to the end of 1973.