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Biological polymers such as nucleic acids and proteins are ubiquitous in living systems, but their ability to address problems beyond those found in nature is constrained by factors such as chemical or biological instability, limited building-block functionality, bioavailability, and immunogenicity. In principle, sequence-defined synthetic polymers based on nonbiological monomers and backbones might overcome these constraints; however, identifying the sequence of a synthetic polymer that possesses a specific desired functional property remains a major challenge. Molecular evolution can rapidly generate functional polymers but requires a means of translating amplifiable templates such as nucleic acids into the polymer being evolved. This review covers recent advances in the enzymatic and nonenzymatic templated polymerization of nonnatural polymers and their potential applications in the directed evolution of sequence-defined synthetic polymers.