This is joint work in progress with Dugald Macpherson (Leeds). The point is to develop a model theory for classes of finite structures that bears a similarity to contemporary model theory for classes of infinite structures, in which some notion of dimension typically plays a central role. We thus refer to the classes we study as, "classes of finite structures with dimension and measure." Our definition of dimension and measure is directly inspired by the paper, "Definable sets over finite fields," by Chatzidakis, van den Dries, and Macintyre (Crelle 1992). Only the most basic familiarity with model theory will be supposed.