In this article we review and summarize salient features of the mechanism of transcription by eukaryotic RNA polymerase III, focusing a considerable part of the account on the work of our laboratory, dealing with Saccharomyces cerevisiae, but emphasizing properties that are common to all eukaryotic RNA polymerases. The following topics are briefly discussed: promoter structure; transcription factors; internal structure of transcription complexes and DNA-protein interactions; U6 genes; promoter opening and RNA chain elongation; hydrolytic RNA retraction.