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Although natural selection may be the most powerful force in the biological world, it is not all powerful. As a consequence, many aspects of evolution of the molecular level can only be explained by the inability of natural selection to operate. This general principle provides an overarching explanation for the evolutionary diversification of genome architecture, and also appears to extend to multiple higher-level features of cells, as most clearly demonstrated with observations on replication fidelity. Understanding the mechanisms of evolution and the degree to which phylogenetic generalities exist requires information on the rate at which mutations arise and their effects at the molecular and phenotypic levels. Information on spontaneous mutations obtained from whole-genome sequencing of mutation-accumulation lines implies an inverse scaling of the mutation rate (per nucleotide site) with the effective population size of a species. This pattern is thought to arise naturally as natural selection pushes the mutation rate down to a lower limit set by the power of random genetic drift rather than by intrinsic molecular limitations on repair mechanisms or by selection for an optimum mutation rate. If this drift-barrier hypothesis is correct, the population-genetic environment imposes a fundamental constraint on the level of perfection that can be achieved by any adaptation at the molecular and phenotypic level. Additional examples to presented will draw from recent observations on the transcription-error rate, the evolution of the oligomeric states of proteins, and the bioenergetic costs of maintaining and expressing genes.