This section is intended as an introduction to, and apology for, multistage models. Such models occupy a curious position in the world of cancer research. It seems likely that multistage processes underlie the generation of a large majority of human cancers (the carcinomas), yet most research workers do not have any real interest in discussing the various alternative multistage models that attempt to describe these processes. These research workers can't all be wrong, so what is wrong with multistage models? The trouble is, I suppose, that the processes usually invoked are, in principle, extremely difficult to observe (phenotypically silent changes in a few scattered somatic cells), and that very similar predictions for the few things we can actually observe may follow from mathematical elaboration of various very different multistage models (each with its own implausible chalones, feedback mechanisms, exact numbers of "stages" to be progressed through, mutations, epigenetic switches, clonal growth rates, etc.).However, laboratory investigations of various completely different aspects of the processes of cancer induction (viruses, mutagens, host control mechanisms, etc.) are already well established, and new lines of investigation, particularly of "cocarcinogenic" processes, may yet emerge. No single process is likely to be the whole truth, and we must hope that some grand synthesis of the known processes will eventually be put together which will describe all the essential features of human cancer induction. Although the eventual synthesis is not yet in sight, multistage models should at present be thought about to some extent, and their general features should be common knowledge, as the general framework of this eventual synthesis will (at least for that 90% of fatal human cancers which are carcinomas) almost certainly be some kind of multistage model. Moreover, despite all their present uncertainties, multistage models for carcinoma induction have already offered plausible answers to various questions concerning monoclonality, dose-response relationships under conditions of regular exposure, hypothetical "threshold" doses, the synergistic effects of different carcinogens, the role of luck, and, last but not least, the connection between cancer and aging. (This latter point has been only partly resolved: multistage models give a very natural explanation for cancer being a hundred times rarer among young adults than among the elderly, but no plausible explanation has yet been offered for the fact that the risk of cancer in old age is not vastly different in species with very different life-spans.)In the next three sections, I will recapitulate a few formulae that can emerge from certain multistage models.(Readers who don't like even simple algebra can pass over