Recent advances in viral metagenomics have enabled the rapid discovery of an unprecedented catalogue of phages in numerous environments, from the human gut to the deep ocean. Although these advances have expanded our understanding of phage genomic diversity, they also revealed that we have only scratched the surface in the discovery of novel viruses. Yet, despite the remarkable diversity of phages at the nucleotide sequence level, the structural proteins that form viral particles show strong similarities and conservation. Phages are uniquely interconnected from an evolutionary perspective and undergo multiple events of genetic exchange in response to the selective pressure of their hosts, which drives their diversity. In this Review, we explore phage diversity at the structural, genomic and community levels as well as the complex evolutionary relationships between phages, moulded by the mosaicity of their genomes.