The theory that an RNA world played a pivotal role in life's evolutionary past has prompted investigations into the scope of RNA catalysis. These efforts have attempted to demonstrate the plausibility of an RNA-based genetic system, which would require RNA molecules that catalyze their own replication. The mechanistic features of modern protein polymerases have been used to guide the laboratory evolution of catalytic RNAs (ribozymes) that exhibit polymerase-like activity. Ribozymes have been developed that recognize a primer-template complex in a general way and catalyze the template-directed polymerization of mononucleotides. These experiments demonstrate that RNA replicase behavior is likely within the catalytic repertoire of RNA, although many obstacles remain to be overcome in order to demonstrate that RNA can catalyze its own replication in a manner that could have sustained a genetic system on the early Earth.