Viruses contain three common types of packaged genomes; double-stranded DNA (dsDNA), RNA (mostly single and occasionally double stranded) and single-stranded DNA (ssDNA). There are relatively straightforward explanations for the prevalence of viruses with dsDNA and RNA genomes, but the evolutionary basis for the apparent success of ssDNA viruses is less clear. The recent discovery of four ssDNA virus genomes that appear to have been formed by recombination between co-infecting RNA and ssDNA viruses, together with the high mutation rate of ssDNA viruses provide possible explanations. RNA-DNA recombination allows ssDNA viruses to access much broader sequence space than through nucleotide substitution and DNA-DNA recombination alone. Multiple non-exclusive mechanisms, all due to the unique replication of ssDNA viruses, are proposed for this unusual RNA capture. RNA capture provides an explanation for the evolutionary success of the ssDNA viruses and may help elucidate the mystery of integrated RNA viruses in viral and cellular DNA genomes.