Relations between conceptual maps and reality are widespread in mathematics. The nature of mathematics itself can be phrased in those terms. Applied mathematicians build mathematical maps of reality, but they normally call them models. We examine the process of mathematical modelling from several distinct directions, examining how a simplified model (map) can often be more informative than a more realistic one (replicating the territory). Examples discussed include Newtonian and Relativistic gravitation, fluid dynamics and the human visual system. We conclude by asking whether digital technology now makes it possible to produce maps that are the same size as the territory, but are still useful.