How life emerged from inanimate matter is one of the most intriguing questions posed to modern science. Central to this research are experimental attempts to build systems capable of Darwinian evolution. RNA catalysts (ribozymes) are a promising avenue, in line with the RNA world hypothesis whereby RNA pre-dated DNA and proteins. Since evolution in living organisms relies on template-based replication, the identification of a ribozyme capable of replicating itself (an RNA self-replicase) has been a major objective. However, no self-replicase has been identified to date. Alternatively, autocatalytic systems involving multiple RNA species capable of ligation and recombination may enable self-reproduction. However, it remains unclear how evolution could emerge in autocatalytic systems. In this review, we examine how experimentally feasible RNA reactions catalysed by ribozymes could implement the evolutionary properties of variation, heredity and reproduction, and ultimately allow for Darwinian evolution. We propose a gradual path for the emergence of evolution, initially supported by autocatalytic systems leading to the later appearance of RNA replicases.