RNA viruses are responsible for some of the worst pandemics known to mankind, including outbreaks of Influenza, Ebola, and the recent COVID-19. One major challenge in tackling RNA viruses is the fact they are extremely genetically diverse. Nevertheless, they share common features that include their dependence on host cells for replication, and high mutation rates. We set out to search for shared evolutionary characteristics that may aid in gaining a broader understanding of RNA virus evolution, and constructed a phylogeny-based dataset spanning thousands of sequences from diverse single-stranded RNA viruses of animals. Strikingly, we found that the vast majority of these viruses have a skewed nucleotide composition, manifested as adenine rich (A-rich) coding sequences. In order to test whether A-richness is driven by selection or by biased mutation processes, we harnessed the effects of incomplete purifying selection at the tips of virus phylogenies. Our results revealed consistent mutational biases towards U rather than A in genomes of all viruses. In +ssRNA viruses we found that this bias is compensated by selection against U and selection for A, which leads to A-rich genomes. In -ssRNA viruses the genomic mutational bias towards U on the negative strand manifests as A-rich coding sequences, on the positive strand. We investigated possible reasons for the advantage of A-rich sequences including weakened RNA secondary structures, codon usage bias, and selection for a particular amino-acid composition, and conclude that host immune pressures may have led to similar biases in coding sequence composition across very divergent RNA viruses.