The Cdv system is the protein machinery that performs cell division and other membrane-deforming processes in a subset of archaea. Evolutionarily, the system is closely related to the eukaryotic ESCRT machinery, with which it shares many structural and functional similarities. Since its first description 15 years ago, the understanding of the Cdv system progressed rather slowly, but recent discoveries sparked renewed interest and insights. The emerging physical picture appears to be that CdvA acts as a membrane anchor, CdvB as a scaffold that localizes division to the mid-cell position, CdvB1 and CvdB2 as the actual constriction machinery, and CdvC as the ATPase that detaches Cdv proteins from the membrane. This paper provides a comprehensive overview of the research done on Cdv and explains how this relatively understudied machinery acts to perform its cell-division function. Understanding of the Cdv system helps to better grasp the biophysics and evolution of archaea, and furthermore provides new opportunities for the bottom-up building of a divisome for synthetic cells.