Biologically based mathematical models of the process of carcinogenesis are not only an essential part of a rational approach to quantitative cancer risk assessment, but also raise fundamental questions about the nature of the events leading to malignancy. In this paper two such models are reviewed. The first is the multistage model proposed by Armitage and Doll in the 1950s. The larger part of the paper is devoted to a discussion of the two-mutation model proposed by Moolgavkar and colleagues. This model is a generalization of the idea of recessive oncogenesis proposed by Knudson, and has been shown to be consistent with a large body of epidemiological and experimental data. The usefulness of the model is illustrated by analysis of a large experimental data set in which rats exposed to radon develop malignant lung tumors.