Embryonic, adult, artificially reprogrammed, and cancer…- there are various types of cells associated with stemness. Do they have something fundamental in common? Are we applying a common name to very different entities? In this review, we will revisit the characteristics that define 'pluripotency', the main property of stem cells (SCs). For each main type of physiological (embryonic and adult) or synthetic (induced pluripotent) SCs, markers and functional behavior in vitro and in vivo will be described. We will review the pioneering work that has led to obtaining human SC lines, together with the problems that have arisen, both in a biological context (DNA alterations, heterogeneity, tumors, and immunogenicity) and with regard to ethical concerns. Such problems have led to proposals for new operative procedures for growing human SCs of sufficiently high quality for use as models of disease and in human therapy. Finally, we will review the data from the first clinical trials to use various types of SCs.