In tumours, a significant fraction of neoplastic cells are engaged in the cell cycle (growth fraction) and are therefore targets for radiation therapy and chemotherapy. Unfortunately, in most disseminated cancers, such treatments cannot lead to complete cure. Many different mechanisms have been described to explain this resistance. The hypothesis of the existence of "cancer stem cells "has been recently proposed. Indeed, the tumour would contain a small subpopulation of cancer cells displaying the phenotypical characteristics of multipotential stem cells. Since such cells display different signalling pathways compared with more differentiated cells, this might explain at least partially the resistance to treatments. Chronic myeloid leukaemia is a good model in favour of cancer stem cells, but the presence of such cells in all types of cancers is still a matter of debate. Several questions emerge: is the multipotential stem cell, the cell of origin of cancer? What is the relevance of the cancer stem cell paradigm for understanding cancer cell biology and to envision new therapeutic, hopefully curative, therapies? The case of chronic myeloid leukaemia is used to discuss these questions.