Most tooth‐bearing non‐mammalian vertebrates have the capacity to replace their teeth throughout life. This capacity was lost in mammals, which replace their teeth only once at most. Not surprisingly, continuous tooth replacement has attracted much attention. Classical morphological studies (e.g. to analyse patterns of replacement) are now being complemented by molecular studies that investigate the expression of genes involved in tooth formation. This review focuses on ray‐finned fish (actinopterygians), which have teeth often distributed throughout the mouth and pharynx, and more specifically on teleost fish, the largest group of extant vertebrates. First we highlight the diversity in tooth distribution and in tooth replacement patterns. Replacement tooth formation can start from a distinct (usually discontinuous and transient) dental lamina, but also in the absence of a successional lamina, e.g. from the surface epithelium of the oropharynx or from the outer dental epithelium of a predecessor tooth. The relationship of a replacement tooth to its predecessor is closely related to whether replacement is the result of a prepattern or occurs on demand. As replacement teeth do not necessarily have the same molecular signature as first‐generation teeth, the question of the actual trigger for tooth replacement is discussed. Much emphasis has been laid in the past on the potential role of epithelial stem cells in initiating tooth replacement. The outcome of such studies has been equivocal, possibly related to the taxa investigated, and the permanent or transient nature of the dental lamina. Alternatively, replacement may result from local proliferation of undifferentiated progenitors, stimulated by hitherto unknown, perhaps mesenchymal, factors. So far, the role of the neurovascular link in continuous tooth replacement has been poorly investigated, despite the presence of a rich vascularisation surrounding actinopterygian (as well as chondrichthyan) teeth and despite a complete arrest of tooth replacement after nerve resection. Lastly, tooth replacement is possibly co‐opted as a process to expand the number of teeth in a dentition ontogenetically whilst conserving features of the primary dentition. That neither a dental lamina, nor stem cells appear to be required for tooth replacement places teleosts in an advantageous position as models for tooth regeneration in humans, where the dental lamina regresses and epithelial stem cells are considered lost.