The objective of this paper is to review the use, in mutagenesis, of various mathematical models to describe the dose-response relationship and to try to identify thresholds. It is often taken as axiomatic that genotoxic carcinogens could damage DNA at any level of exposure, leading to a mutation, and that this could ultimately result in tumour development. This has led to the assumption that for genotoxic chemicals, there is no discernible threshold. This assumption is increasingly being challenged in the case of aneugens. The distinction between 'absolute' and 'pragmatic' thresholds is made and the difficulties in determining 'absolute' thresholds using hypothesis testing approaches are described. The potential of approaches, based upon estimation rather than statistical significance for the characterization of dose-response relationships, is stressed. The achievement of a good fit of a mathematical model to experimental data is not proof that the mechanism supposedly underlying this model is operating. It has been argued, in the case of genotoxic chemicals, that any effects produced by a genotoxic chemical which augments that producing a background incidence in unexposed individuals will lead to a dose-response relationship that is non-thresholded and is linear at low doses. The assumptions underlying this presumpti