In this review we evaluate evidence for three different hypotheses that explain how the corneal epithelium is maintained. The limbal epithelial stem cell (LESC) hypothesis is most widely accepted. This proposes that stem cells in the basal layer of the limbal epithelium, at the periphery of the cornea, maintain themselves and also produce transient (or transit) amplifying cells (TACs). TACs then move centripetally to the centre of the cornea in the basal layer of the corneal epithelium and also replenish cells in the overlying suprabasal layers. The LESCs maintain the corneal epithelium during normal homeostasis and become more active to repair significant wounds. Second, the corneal epithelial stem cell (CESC) hypothesis postulates that, during normal homeostasis, stem cells distributed throughout the basal corneal epithelium, maintain the tissue. According to this hypothesis, LESCs are present in the limbus but are only active during wound healing. We also consider a third possibility, that the corneal epithelium is maintained during normal homeostasis by proliferation of basal corneal epithelial cells without any input from stem cells. After reviewing the published evidence, we conclude that the LESC and CESC hypotheses are consistent with more of the evidence than the third hypothesis, so we do not consider this further. The LESC and CESC hypotheses each have difficulty accounting for one main type of evidence so we evaluate the two key lines of evidence that discriminate between them. Finally, we discuss how lineage-tracing experiments have begun to resolve the debate in favour of the LESC hypothesis. Nevertheless, it also seems likely that some basal corneal epithelial cells can act as long-term progenitors if limbal stem cell function is compromised. Thus, this aspect of the CESC hypothesis may have a lasting impact on our understanding of corneal epithelial maintenance, even if it is eventually shown that stem cells are restricted to the limbus as proposed by the LESC hypothesis.