Human papillomavirus (HPV) infections are a major human health problem; they are the cause of recurrent benign warts and of several cancers of the anogenital tract and head and neck region. Although there are two prophylactic HPV vaccines that could, if used universally, prevent as many as two-thirds of HPV-induced cancers, as well as several cytotoxic and immunomodulatory agents for localized treatment of infections, there are currently no HPV antiviral drugs in our arsenal of therapeutic agents. This review examines the status of past and ongoing research into the development of HPV antivirals, focused primarily upon approaches targeting the replication of the viral genome. The only HPV enzyme, E1, is a DNA helicase that interfaces with the cellular DNA replication machinery to replicate the HPV genome. To date, searches for small molecule inhibitors of E1 for use as antivirals have met with limited success. The lack of other viral enzymes has meant that the search for antivirals has shifted to a large degree to the modulation of protein-protein interactions. There has been some success in identifying small molecule inhibitors targeting interactions between HPV proteins but with activity against a small subset of viral types only. As noted in this review, it is thought that targeting E1 interactions with cellular replication proteins may provide inhibitors with broader activity against multiple HPV types. Herein, we outline the steps in HPV DNA replication and discuss those that appear to provide the most advantageous targets for the development of anti-HPV therapeutics.