The theory of the clonal origin of cancer states that a tumour arises from one cell that acquires mutation(s) leading to the malignant phenotype. It is the current belief that many of these mutations give a fitness advantage to the mutant population allowing it to expand, eventually leading to disease. However, mutations that lead to such a clonal expansion need not give a fitness advantage and may in fact be neutral--or almost neutral--with respect to fitness. Such mutant clones can be eliminated or expand stochastically, leading to a malignant phenotype (disease). Mutations in haematopoietic stem cells give rise to diseases such as chronic myeloid leukaemia (CML) and paroxysmal nocturnal haemoglobinuria (PNH). Although neutral drift often leads to clonal extinction, disease is still possible, and in this case, it has important implications both for the incidence of disease and for therapy, as it may be more difficult to eliminate neutral mutations with therapy. We illustrate the consequences of such dynamics, using CML and PNH as examples. These considerations have implications for many other tumours as well.