Planarians are members of the Platyhelminthes (flatworms). These animals have evolved a remarkable stem cell system. A single pluripotent adult stem cell type ("neoblast") gives rise to the entire range of cell types and organs in the planarian body plan, including a brain, digestive-, excretory-, sensory- and reproductive systems. Neoblasts are abundantly present throughout the mesenchyme and divide continuously. The resulting stream of progenitors and turnover of differentiated cells drive the rapid self-renewal of the entire animal within a matter of weeks. Planarians grow and literally de-grow ("shrink") by the food supply-dependent adjustment of organismal turnover rates, scaling body plan proportions over as much as a 50-fold size range. Their dynamic body architecture further allows astonishing regenerative abilities, including the regeneration of complete and perfectly proportioned animals even from tiny tissue remnants. Planarians as an experimental system, therefore, provide unique opportunities for addressing a spectrum of current problems in stem cell research, including the evolutionary conservation of pluripotency, the dynamic organization of differentiation lineages and the mechanisms underlying organismal stem cell homeostasis. The first part of this review focuses on the molecular biology of neoblasts as pluripotent stem cells. The second part examines the fascinating mechanistic and conceptual challenges posed by a stem cell system that epitomizes a universal design principle of biological systems: the dynamic steady state.