Contrary to their host cells, many viruses contain RNA as genetic material and hence encode an RNA-dependent RNA polymerase to replicate their genomes. This review discusses the present status of our knowledge on the structure of these enzymes and the mechanisms of RNA replication. The simplest viruses encode only the catalytic subunit of the replication complex, but other viruses also contribute a variable number of ancillary factors. These and other factors provided by the host cell play roles in the specificity and affinity of template recognition and the assembly of the replication complex. Usually, these host factors are involved in protein synthesis or RNA modification in the host cell, but they play roles in remodeling RNA-RNA, RNA-protein, and protein-protein interactions during virus RNA replication. Furthermore, viruses take advantage of and modify previous cell structural elements, frequently membrane vesicles, for the formation of RNA replication complexes.