Organoids are three-dimensional structures that are derived from the self-organization of stem cells as they differentiate in vitro. The plasticity of stem cells is one of the major criteria for generating organoids most similar to the tissue structures they intend to mimic. Stem cells are cells with unique properties of self-renewal and differentiation. Depending on their origin, a distinction is made between pluripotent (embryonic) stem cells (PSCs), adult (or tissue) stem cells (ASCs), and those obtained by somatic reprogramming, so-called induced pluripotent stem cells (iPSCs). While most data since the 1980s have been acquired in the mouse model, and then from the late 1990s in humans, the process of somatic reprogammation has revolutionized the field of stem cell research. For domestic animals, numerous attempts have been made to obtain PSCs and iPSCs, an approach that makes it possible to omit the use of embryos to derive the cells. Even if the plasticity of the cells obtained is not always optimal, the recent progress in obtaining reprogrammed cells is encouraging. Along with PSCs and iPSCs, many organoid derivations in animal species are currently obtained from ASCs. In this study, we present state-of-the-art stem cell research according to their origins in the various animal models developed.