This review discusses the similarities and differences in the processes of constitutive ("adult") neurogenesis, which have arisen in the course of evolution in higher and lower vertebrates. The data on the features of the regenerative potential in various animal species manifested under conditions of damage to the CNS and the possibilities for restoration due to the regeneration capabilities are analyzed. The influence of a number of factors on the formation of newly created neurons in the adult brain, namely of the effects of different stresses and nutritional status, those of physical exercise, socio-sexual interactions, size of the brain, and aging process, are described. The influence of CNS damage on the processes of neurogenesis at the cellular and molecular levels is discussed in detail. Activation of the potential of structural plasticity of the CNS may open up new opportunities for the development of therapeutic approaches for the treatment of neurodegenerative (in particular age-related) diseases and consequences of the traumatic brain injuries. Mechanisms of regulation of the development of neuronal stem cells, characteristics of the neurogenic zones, processes of migration of newly formed neurons, and their integration into the existing neural networks in the adult brain are analyzed. Special attention is paid to the contradictions in recent publications in the most authoritative scientific journals, which initiated a large number of discussions on whether constitutive neurogenesis does occur in the CNS structures (first of all, in the hippocampus) of adult humans.