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Since 2000, when the first Markov model of the natural history of HPV infection and cervical cancer was published, there has been much progress in the mathematical and statistical modelling of cervical carcinogenesis with a view to project the impact of new technologies in screening and the role of HPV vaccination. Static Markov models were gradually replaced by dynamic models which simulated the risk of acquisition of individual HPV genotypes and their downstream consequences in risk of cervical precancerous lesions and from that to cancer. The role of natural immunity, herd immunity, hypothetical strategies for HPV vaccination, and algorithms for cervical cancer screening could then be overlaid into ever more sophisticated mathematical models. Economical inputs related to costs, utilities, and discounting based on the expected contribution of preventive interventions were part of the models that generated the bulk of cost-effectiveness analyses that have informed policy making in the control of HPV-associated cancers. These mathematical models have to rely on a credible knowledge base of epidemiological data on HPV transmission and cervical carcinogenesis. The author will summarize the state of that knowledge and on how it has come to assist progress in this area.