In a quantum system with an infinite number of degrees of freedom, loop corrections may break symmetries of the original Lagrangian. This phenomenon, a ``quantum anomaly'', arises from the need for a ``regularization'': a supplemental definition of the theory in the ultraviolet region. A supplemental definition of this sort unavoidably runs into a contradiction with certain symmetries of the classical theory. In particular, it causes a nonconservation of corresponding Noether currents. Reasons for the appearance of anomalies and their place in the structure of modern field-theory models are discussed in this review. An emphasis is placed on anomalies in the internal currents of gauge theories. These anomalies may disrupt the invariance under infinitesimal or global gauge transformations, with the result that the theory is no longer self-consistent. The condition which must be met for the cancellation of internal anomalies severely restricts the composition of fields and the choice of interaction in realistic models. Methods for calculating anomalies are discussed in detail. Emphasis is placed on the nonconservation of axial and chiral fermion currents. The hierarchy of anomalies is introduced. A special section is devoted to global anomalies, in particular, Witten's SU(2) anomaly and a corresponding phenomenon in odd-dimensional Yang--Mills theories.