One out of 10 cancers is estimated to arise from infections by a handful of oncogenic viruses. These infectious cancers constitute an opportunity for primary prevention through immunization against the viral infection, for early screening through molecular detection of the infectious agent, and potentially for specific treatments, by targeting the virus as a marker of cancer cells. Accomplishing these objectives will require a detailed understanding of the natural history of infections, the mechanisms by which the viruses contribute to disease, the mutual adaptation of viruses and hosts, and the possible viral evolution in the absence and in the presence of the public health interventions conceived to target them. This issue showcases the current developments in experimental tissue-like and animal systems, mathematical models and evolutionary approaches to understand DNA oncoviruses. Our global aim is to provide proximate explanations to the present-day interface and interactions between virus and host, as well as ultimate explanations about the adaptive value of these interactions and about the evolutionary pathways that have led to the current malignant phenotype of oncoviral infections. This article is part of the theme issue ‘Silent cancer agents: multi-disciplinary modelling of human DNA oncoviruses’.