Human tissues acquire somatic mitochondrial DNA (mtDNA) mutations with age. Very high levels of specific mtDNA mutations accumulate within individual cells, causing a defect of mitochondrial oxidative metabolism. This is a fundamental property of nondividing tissues, but it is not known how it comes about. To explore this problem, we developed a model of mtDNA replication within single human cells. Using this model, we show that relaxed replication of mtDNA alone can lead, through random genetic drift, to the clonal expansion of single mutant events during human life. Significant expansions primarily develop from mutations acquired during a critical period in childhood or early adult life.