Abstract Aim The aims of this study are to resolve terminological confusion around different types of species–area relationships (SARs) and their delimitation from species sampling relationships (SSRs), to provide a comprehensive overview of models and analytical methods for SARs, to evaluate these theoretically and empirically, and to suggest a more consistent approach for the treatment of species–area data. Location Curonian Spit in north-west Russia and archipelagos world-wide. Methods First, I review various typologies for SARs and SSRs as well as mathematical models, fitting procedures and goodness-of-fit measures applied to SARs. This results in a list of 23 function types, which are applicable both for untransformed (S) and for log-transformed (log S) species richness. Then, example data sets for nested plots in continuous vegetation (n =14) and islands (n =6) are fitted to a selection of 12 function types (linear, power, logarithmic, saturation, sigmoid) both for S and for log S. The suitability of these models is assessed with Akaike’s information criterion for S and log S, and with a newly proposed metric that addresses extrapolation capability. Results SARs, which provide species numbers for different areas and have no upper asymptote, must be distinguished from SSRs, which approach the species richness of one single area asymptotically. Among SARs, nested plots in continuous ecosystems, non-nested plots in continuous ecosystems, and isolates can be distinguished. For the SARs of the empirical data sets, the normal and quadratic power functions as well as two of the sigmoid functions (Lomolino, cumulative beta-P) generally performed well. The normal power function (fitted for S) was particularly suitable for predicting richness values over ten-fold increases in area. Linear, logarithmic, convex saturation and logistic functions generally were inappropriate. However, the two sigmoid models produced unstable results with arbitrary parameter estimates, and the quadratic power function resulted in decreasing richness values for large areas. Main conclusions Based on theoretical considerations and empirical results, I suggest that the power law should be used to describe and compare any type of SAR while at the same time testing whether the exponent z changes with spatial scale. In addition, one should be aware that power-law parameters are significantly influenced by methodology.