Viral diversity and evolution play a central role in processes such as disease emergence, vaccine failure, drug resistance, and virulence. However, significant challenges remain to better understand and manage these processes. Here, we discuss five of these challenges. These include improving our ability to predict viral evolution, developing more relevant experimental evolutionary systems, integrating viral dynamics and evolution at different scales, systematic appraisal of the virosphere, and deepening our understanding of virus-virus interactions. Intensifying future research on these areas should improve our ability to combat viral diseases, as well as to exploit viral diversity and evolution for biotechnological purposes.