An analysis is made of the results of EPR experiments in superconducting and nonsuperconducting samples of YBa$_2$Cu$_3$O$_{7-\delta}$, including samples irradiated by neutrons and samples with various admixed impurities. It is shown that the observed resonance signal in superconducting samples is due to impurity phases and that the diversity of detected signals from oxygen-deficient samples is probably due to polymorphism in the distribution of oxygen atoms. The results of investigations of the so-called low-field signal, which is due to the presence of a network of Josephson weak links, are discussed in detail for both ceramics and single crystals. The findings of EPR studies in high-temperature superconductors are summarized, and possible directions for further experiments are discussed.