The midbody is a singular organelle formed between daughter cells during cytokinesis and required for their final separation. Midbodies persist in cells long after division as midbody derivatives (MB^ds), but their fate is unclear. Here we show that MB^ds are inherited asymmetrically by the daughter cell with the older centrosome. They selectively accumulate in stem cells, induced pluripotent stem cells and potential cancer ‘stem cells’ in vivo and in vitro. MB^d loss accompanies stem-cell differentiation, and involves autophagic degradation mediated by binding of the autophagic receptor NBR1 to the midbody protein CEP55. Differentiating cells and normal dividing cells do not accumulate MB^ds and possess high autophagic activity. Stem cells and cancer cells accumulate MB^ds by evading autophagosome encapsulation and exhibit low autophagic activity. MB^d enrichment enhances reprogramming to induced pluripotent stem cells and increases the in vitro tumorigenicity of cancer cells. These results indicate unexpected roles for MB^ds in stem cells and cancer ‘stem cells’.