Many philosophical schools of mathematics reject the notion of infinity, and restrict attention to finite structures and objects. At first sight it would seem that they could not deal at all with classical geometry, since its usual entities (segments, lines, circles) all consist of infinitely many points. To bypass the problem, modern mathematics has developed a number of finite versions of Euclidean and non Euclidean geometries, which raise many subtle and difficult problems, the solutions of some of which have even led to a couple of Fields medals.