The past few years have seen the rather remarkable application of number theory (diophantine approximation theory and p-adic analysis) to the computation and classification of homogeneous integrable systems. The basic tool is the symbolic method of Gelfand and Dikii, which translates differential expressions into polynomials. In this talk I'll review these developments, both in the commutative and the noncommutative case, in such a way that a nonspecialist will be able to follow it. In particular the recent classification of second order two component systems will be discussed.