The Epstein-Barr virus (EBV), one of the most common in the human population, is capable of lifelong persistence in resting memory B-cells, in T-cells in case of type 2 EBV, and in some undifferentiated epithelial cells. In most people, EBV persistence is not accompanied by significant symptoms, but frequent virus activations are associated with the increased risks of severe diseases, such as chronic active Epstein-Barr virus infection, hemophagocytic lymphohistiocytosis, multiple sclerosis, systemic lupus erythematosus, gastric and nasopharyngeal carcinomas, and a variety of T- and B-cell lymphomas. Therefore, the molecular viral and host cell processes during asymptomatic or low-symptom EBV persistence are of great interest. This review describes the behavior of the viral DNA in an infected cell and the forms of its existence (linear, circular episome, chromosomally integrated forms), as well as methods of EBV genome copying. Two closely related cycles of viral reproduction are considered. Lytic activation is unfavorable for the survival of a particular viral genome in the cell, and may be a result of differentiation of a latently infected cell, or the arrival of stress signals due to adverse extracellular conditions. The EBV has a large number of adaptive mechanisms for limiting lytic reactivation and reducing hostility of host immune cells. Understanding the molecular aspects of EBV persistence will help in the future develop more effective targeted drugs for the treatment of both viral infection and associated diseases.