Genetic recombination in positive-strand RNA viruses is a significant evolutionary mechanism that drives the creation of viral diversity by the formation of novel chimaeric genomes. The process and its consequences, for example the generation of viruses with novel phenotypes, has historically been studied by analysis of the end products. More recently, with an appreciation that there are both replicative and non-replicative mechanisms at work, and with new approaches and techniques to analyse intermediate products, the viral and cellular factors that influence the process are becoming understood. The major influence on replicative recombination is the fidelity of viral polymerase, although RNA structures and sequences may also have an impact. In replicative recombination the viral polymerase is necessary and sufficient, although roles for other viral or cellular proteins may exist. In contrast, non-replicative recombination appears to be mediated solely by cellular components. Despite these insights, the relative importance of replicative and non-replicative mechanisms is not clear. Using single-stranded positive-sense RNA viruses as exemplars, we review the current state of understanding of the processes and consequences of recombination.