Astroglial cells are the most frequent cell type in the adult mammalian brain, and the number and range of their diverse functions are still increasing. One of their most striking roles is their function as adult neural stem cells and contribution to neurogenesis. This chapter discusses first the role of the ubiquitous glial cell type in the developing nervous system, the radial glial cells. Radial glial cells share several features with neuroepithelial cells, but also with astrocytes in the mature brain, which led to the name "radial glia." At the end of neurogenesis in the mammalian brain, radial glial cells disappear, and a subset of them transforms into astroglial cells. Interestingly, only some astrocytes maintain their neurogenic potential and continue to generate neurons throughout life. We discuss the current knowledge about the differences between the adult astroglial cells that remain neurogenic and act as neural stem cells and the majority of other astroglial cells that have apparently lost the capacity to generate neurons. Additionally, we review the changes in glial cells upon brain lesion, their dedifferentiation and recapitulation of radial glial properties, and the conditions under which reactive glia may reinitiate some neurogenic potential. Given that the astroglial cells are not only the most frequent cell type in an adult mammalian brain, but also the key cell type in the wound reaction of the brain to injury, it is essential to further understand their heterogeneity and molecular specification, with the final aim of using this unique source for neuronal replacement. Therefore, one of the key advances in the field of neurobiology is the discovery that astroglial cells can generate neurons not only during development, but also throughout adult life and potentially even after brain lesion.