Recent studies have identified primitive, malignant stem cells which have entered the G0-phase of the cell cycle to become 'quiescent' and which are present, in small numbers, in all chronic myeloid leukaemia (CML) patients. These cells have attracted intense scrutiny because they are proving exceptionally refractory to attempts to kill them, in vitro, using imatinib mesylate, the current first-line therapy for CML, or conventional chemotherapeutic agents, such as cytosine arabinoside. This insensitivity, or resistance, to drug treatment is ominous and has important implications for the clinical management of CML, particularly with regard to relapse following an imatinib-induced remission. In this review, we consider the known properties of this cell population, including recent evidence which suggests that transcription of BCR-ABL occurs at an exceptionally high level in these cells despite them having only a single copy of the oncogene. We also discuss possible alternative, Bcr-Abl-independent, mechanisms for the insensitivity of these cells to agents which promote apoptosis, including the putative role of transporter proteins in causing abnormal drug influx or efflux.