The review sets forth the main results of theoretical and experimental studies of the radiation emitted by relativistic charged particles in undulators. The physical basis of the action of undulators is described. The main properties of undulator radiation and their relation to the trajectory of the particles in the undulator are discussed. Quantitative relations are given for the spectral, angular, and polarization characteristics of the radiation by an individual particle. The influence of the electron-beam parameters on the properties of the radiation are considered. The relation between the size of the diaphragm and the spectral characteristics of the radiation selected by it is analyzed. Some fundamental aspects and the results of recent experiments on generation of coherent undulator radiation are presented, including radiation from free electron lasers. Various types and features of undulators are described, and also characteristics of existing sources of undulator radiation in synchrotrons and storage rings. Several methods of experimental investigation and results obtained therein are discussed.