Mitochondria constitute the major energy-producing compartment of the eukaryotic cell. These organelles contain many molecules of DNA that contribute only a handful of proteins required for energy production. Mutations in the DNA of mitochondria were identified as a cause of human disease a quarter of a century ago, and they have subsequently been implicated in ageing. The process whereby deleterious variants come to dominate a cell, tissue or human is the subject of debate. It is likely to involve multiple, often competing, factors, as selection pressures on mitochondrial DNA can be both indirect and intermittent, and are subjected to rapid change. Here, we assess the different models and the prospects for preventing the accumulation of deleterious mitochondrial DNA variants with time.