Plants continuously generate new tissues and organs throughout their life cycle, due to the activity of populations of specialized tissues containing stem cells called meristems. The shoot apical meristem (SAM) generates all the aboveground organs of the plant, including leaves and flowers, and plays a key role in plant survival and reproduction. Organ production at the SAM occurs following precise spatio-temporal patterns known as phyllotaxis. Because of the regularity of these patterns, phyllotaxis has been the subject of investigations from biologists, physicists, and mathematicians for several centuries. Both experimental and theoretical works have led to the idea that phyllotaxis results from a self-organizing process in the meristem via long-distance interactions between organs. In recent years, the phytohormone auxin has emerged not only as the central regulator of organogenesis at the SAM, but also as a major determinant of the self-organizing properties of phyllotaxis. Here, we discuss both the experimental and theoretical evidence for the implication of auxin in the control of organogenesis and self-organization of the SAM. We highlight how several layers of control acting at different scales contribute together to the function of the auxin signal in SAM dynamics. We also indicate a role for mechanical forces in the development of the SAM, supported by recent interdisciplinary studies.