Although chemotherapy of tumours has scored successes, drug resistance remains the major cause of death of cancer patients. Initial treatment often leaves residual disease, from which the tumour regrows. Eventually, most tumours become resistant to all available chemotherapy. I call this pan-resistance to distinguish it from multi-drug resistance, usually describing resistance caused by upregulation of drug transporters, such as P-glycoprotein. In this review, I discuss mechanisms proposed to explain both residual disease and pan-resistance. Although plausible explanations are at hand for residual disease, pan-resistance is still a mystery. My conclusion is that it is time for a major effort to solve this mystery using the new genetically modified mouse tumour models that produce real tumours resembling cancer in human patients.