Since their discovery during the last century, the asteroids, or minor planets, have remained one of the principal anomalies in the solar system. The purpose of the present work is to construct a theoretical model of the asteroid system, from which something of the origin and evolution of the asteroids might be learned. First, the history of the discovery of the asteroids is outlined together with their major physical properties, and the probable connexion between asteroids and meteorites is discussed. In seeking an explanation of the asteroid system the early theory of a single disrupted planet must be dismissed. Yet of the various hypotheses put forward, a process of fragmentation seems by far the most likely. Recent theories based on fragmentation are therefore reviewed. Our own model is then presented. In describing the results of collisional fragmentation it is easiest to treat the distribution of asteroid masses. Accordingly, equations describing the change with time of the asteroid mass distribution have been derived. Solutions, both analytical and numerical, are presented here, and their consequences discussed. Special attention is given to the important subsidiary problem of the break-up of individual asteroids.