Dynamical Galois groups are profinite groups that are constructed via iterations of rational functions. They are intimately connected with, and in a way they are a wild generalization of, Galois representations on Tate modules of elliptic curves. The problem of determining which rational functions yield abelian dynamical Galois groups has attracted quite some attention in recent years; in this talk I will survey on what is known about this problem, and explain how to solve certain instances of it over number fields and over function fields. This is based on joint works with P. Ingram, A. Ostafe, C. Pagano and U. Zannier.