After a brief introduction in which the concepts of the magnetic and electric dipole moments of particles are introduced and questions are discussed associated with the discrete $P$ and $T$ transformations, the main part of the paper follows which consists of two sections. In the first of these the validity of quantum electrodynamics is analyzed on the basis of measurements of the anomalous magnetic moments of electrons and muons. Special emphasis is placed on experimental methods of measurement and on their historical development. The most detailed description is provided of three recent experiments: 1) measurement of $(g-2)$ for muons at CERN; 2) the spin resonance experiment on single electrons at the University of Washington; 3) the precise comparison of electron and positron magnetic moments carried out using the VEPP-2M storage ring at Novosibirsk. In the next section a review is given of the tests of $CPT$, $CP$, and $T$ invariance from muon decay correlations and lifetime measurements. Finally the present status of experimental searches for an electric dipole moment (the existence of which would violate both $P$ and $T$ invariance) of electrons and muons is discussed.