Abstract: The group SL(2,Z) of two by two matrices with integer entries and determinant 1 has a pleasingly simple definition. Yet its representation theory is quite wild, and at the same time, it is important in many fields of research reaching from quantum theory to number theory. It has uncountably many irreducible representations. On the other hand, as we shall see in this talk, if we restrict to the important class of representations with finite image whose kernel can be described by congruences, the representation theory becomes quite well-behaved, and it is intimately related to the theory of quadratic forms with integral coefficients. In this talk we have, first of all, a closer look at the group SL(2,Z). We shall then describe what is known about its representation theory, and concentrate finally on the representations whose kernel can be described as the set of matrices whose entries satisfy certain congruences. At the end, we shall indicate some applications in other fields, and we shall discuss possible ways to study in a similar way the groups SL(2,O), where O is the ring of integers in a number field.