A mathematical model previously developed for use in computer vision applications is presented as an empirical model for face space. The term appearance space is used to distinguish this from previous models. Appearance space is a linear vector space that is dimensionally optimal, enables us to model and describe any human facial appearance, and possesses characteristics that are plausible for the representation of psychological face space. Randomly sampling from a multivariate distribution for a location in appearance space produces entirely plausible faces, and manipulation of a small set of defining parameters enables the automatic generation of photo-realistic caricatures. The appearance space model leads us to the new concept of nonlinear caricatures, and we show that the accepted linear method for caricature is only a special case of a more general paradigm. Nonlinear methods are also viable, and we present examples of photographic quality caricatures, using a number of different transformation functions. Results of a simple experiment are presented that suggest that nonlinear transformations can accurately capture key aspects of the caricature effect. Finally, we discuss the relationship between appearance space, caricature, and facial distinctiveness. On the basis of our new theoretical framework, we suggest an experimental approach that can yield new evidence for the plausibility of face space and its ability to explain processes of recognition.