Abstract
We consider infinite classes of finite structures, in a fixed first order language, where the sizes of the definable sets have good asymptotic behaviour as the structures become large. The motivating example of a 1-dimensional asymptotic class is the class of finite fields (by a result of Catzidakis, Macintyre and van den Dries). Here we generalise and consider N-dimensional asymptotic classes, and show that they lie within the general context of supersimple theories of finite rank. We provide further examples, develop the general theory of these classes, and (in joint work with Mark Ryten) begin classifying algebraic objects in this setting.