In animals including humans, mutation rates per generation exceed a perceived threshold, and excess mutations increase genetic load. Despite this, animals have survived without extinction. This is a perplexing problem for animal and human genetics, arising at the end of the last century, and to date still does not have a fully satisfactory explanation. Shortly after we proposed the disparity theory of evolution in 1992, the disparity mutagenesis model was proposed, which forms the basis for an explanation for an acceleration of evolution and species survival. This model predicts a significant increase of the mutation threshold values if the fidelity difference in replication between the lagging and leading strands is high enough. When applied to biological evolution, the model predicts that living things, including humans, might overcome the lethal effect of accumulated deleterious mutations and be able to survive. Artificially derived mutator strains of microorganisms, in which an enhanced lagging-strand-biased mutagenesis was introduced, showed unexpectedly high adaptability to severe environments. The implications of the striking behaviors shown by these disparity mutators will be discussed in relation to how living things with high mutation rates can avoid the self-defeating risk of excess mutations.