| Seshadri constants were introduced by Demailly around 1990 and measure the local positivity of a line bundle on a variety. A subtle point is that they are extremely hard to compute in most cases; for instance no example is known where they are irrational. In the talk I will give an account of what is known on K3 surfaces, in particular, on recent work with Concettina Galati where we compute Seshadri constants on K3 surfaces of degrees 4 and 6 in P^4 and P^5, respectively. The case of a quartic surface in P^3 had been previously settled by Thomas Bauer. |