Because exposure to carcinogens cannot be eliminated, it has become important to define exposure levels which are acceptable to society or are irreducible. Quantitative assessment of cancer risk is part of this process. This paper presents a definition of risk and discusses the role of epidemiologic observation in the quantitative assessment of cancer risk, the estimation of risk from epidemiologic data, and the role of animal cancer bioassays in the quantitative assessment of cancer risk. It is emphasized that quantitative risk assessment is inevitably based on multiple assumptions. Where possible, the magnitude of errors associated with these assumptions should be stated, even to the extent of an acknowledgment of complete ignorance.