The evolution of ribonucleic acid (RNA) molecules in the test tube provides a possible way to study evolutionary optimization and adaptation to the environment on time scales accessible to human observers. Diversity of genotypes, however, is prohibitive for a complete experimental recording of the process at the molecular level. The number of RNA sequences and structures is too large to be determined by means of currently available techniques. Computer simulation, on the other hand, is able to handle large numbers of individual sequences and has no major problem with data retrieval. However, it can deal only with simplified relations between genotypes and phenotypes, i.e., RNA sequences and structures, respectively. Based on a coarse-grained notion of structure, as represented by RNA secondary structures, for example, a comprehensive model of evolution has been developed that allows as to follow optimization at full molecular resolution. This model describes the course of in vitro selection experiments, and provides a straightforward explanation for the occurrence of steps observed in evolution. It initiated the development of new mathematical concepts which analyse evolution as a complex process viewed simultaneously in concentration space, sequence space, and shape space.