Seven kinds of virus collectively comprise an important cause of cancer, particularly in less developed countries and for people with damaged immune systems. Discovered over the past 54 years, most of these viruses are common infections of humankind for which malignancy is a rare consequence. Various cofactors affect the complex interaction between virus and host and the likelihood of cancer emerging. Although individual human tumour viruses exert their malignant effects in different ways, there are common features that illuminate mechanisms of oncogenesis more generally, whether or not there is a viral aetiology. This article is part of the themed issue ‘Human oncogenic viruses’.