Just as Darwinian evolution in nature has led to the development of many sophisticated enzymes, Darwinian evolution in vitro has proven to be a powerful approach for obtaining similar results in the laboratory. This review focuses on the development of nucleic acid enzymes starting from a population of random-sequence RNA or DNA molecules. In order to illustrate the principles and practice of in vitro evolution, two especially well-studied categories of catalytic nucleic acid are considered: RNA enzymes that catalyze the template-directed ligation of RNA and DNA enzymes that catalyze the cleavage of RNA. The former reaction, which involves attack of a 2'- or 3'-hydroxyl on the alpha-phosphate of a 5'-triphosphate, is more difficult. It requires a comparatively larger catalytic motif, containing more nucleotides than can be sampled exhaustively within a starting population of random-sequence RNAs. The latter reaction involves deprotonation of the 2'-hydroxyl adjacent to the cleavage site, resulting in cleaved products that bear a 2',3'-cyclic phosphate and 5'-hydroxyl. The difficulty of this reaction, and therefore the complexity of the corresponding DNA enzyme, depends on whether a catalytic cofactor, such as a divalent metal cation or small molecule, is present in the reaction mixture.