The demonstration that ribosomal peptide synthesis is a ribozyme-catalyzed reaction makes it almost certain that there was once an RNA World. The central problem for origin-of-life studies, therefore, is to understand how a protein-free RNA World became established on the primitive Earth. We first review the literature on the prebiotic synthesis of the nucleotides, the nonenzymatic synthesis and copying of polynucleotides, and the selection of ribozyme catalysts of a kind that might have facilitated polynucleotide replication. This leads to a brief outline of the Molecular Biologists' Dream, an optimistic scenario for the origin of the RNA World. In the second part of the review we point out the many unresolved problems presented by the Molecular Biologists' Dream. This in turn leads to a discussion of genetic systems simpler than RNA that might have "invented" RNA. Finally, we review studies of prebiotic membrane formation.