In the mammalian brain, neurogenesis is maintained throughout adulthood primarily in two typical niches, the subgranular zone (SGZ) of the dentate gyrus and the subventricular zone (SVZ) of the lateral ventricles and in other nonclassic neurogenic areas (e.g., the amygdala and striatum). During prenatal and early postnatal development, neural stem cells (NSCs) differentiate into neurons and migrate to appropriate areas such as the olfactory bulb where they integrate into existing neural networks; these phenomena constitute the multistep process of neurogenesis. Alterations in any of these processes impair neurogenesis and may even lead to brain dysfunction, including cognitive impairment and neurodegeneration. Here, we first summarize the main properties of mammalian neurogenic niches to describe the cellular and molecular mechanisms of neurogenesis. Accumulating evidence indicates that neurogenesis plays an integral role in neuronal plasticity in the brain and cognition in the postnatal period. Given that neurogenesis can be highly modulated by a number of extrinsic and intrinsic factors, we discuss the impact of extrinsic (e.g., alcohol) and intrinsic (e.g., hormones) modulators on neurogenesis. Additionally, we provide an overview of the contribution of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection to persistent neurological sequelae such as neurodegeneration, neurogenic defects and accelerated neuronal cell death. Together, our review provides a link between extrinsic/intrinsic factors and neurogenesis and explains the possible mechanisms of abnormal neurogenesis underlying neurological disorders.