Although the species–area relationship (SAR) is commonly presumed to be either a power law or to follow the logarithmic relationship, a large number of other mathematical expressions have been proposed to describe the relationship. These models can be divided into four general categories, distinguishing between asymptotic and non-asymptotic, and between convex upward and sigmoid models (in arithmetic space). The choice of regression model should not be determined by best fit alone; rather, the choice should relate to the purpose of fitting mathematical models to SAR data: either descriptive, explicative or predictive. Therefore, we should choose models that are likely to result from expected ecological patterns. We argue that neither (accumulative) sample-area SARs (saSARs) nor island SARs (ISARs) have upper asymptotes and ISARs may be sigmoid if the smallest islands (finest scales) are included. Amongst the 30 different models we review here, few are non-asymptotic. Both the power model and logarithmic model return convex non-asymptotic curves, whereas the second persistence (P2) model and the quadratic logarithmic model consistently return sigmoid curves without asymptotes. We add the Tjørve-hybrid to this shortlist, as it can be useful when neither the power nor the logarithmic model provides a good fit to saSAR data.