The cellular surveillance network for sensing and repairing damaged DNA prevents an array of human diseases, and when compromised it can lead to genomic instability and cancer. The carefully maintained cellular response to DNA damage is challenged during viral infection, when foreign DNA is introduced into the cell. The battle between virus and host generates a genomic conflict. The host attempts to limit viral infection and protect its genome, while the virus deploys tactics to eliminate, evade, or exploit aspects of the cellular defense. Studying this conflict has revealed that the cellular DNA damage response machinery comprises part of the intrinsic cellular defense against viral infection. In this review we examine recent advances in this emerging field. We identify common themes used by viruses in their attempts to commandeer or circumvent the host cell's DNA repair machinery, and highlight potential outcomes of the conflict for both virus and host.