How complicated the dynamics of a rational map can be? In the talk, we will explain why for large families of rational maps of arbitrary degree their dynamics is no more complicated than the dynamics of complex polynomials, in a very precise sense. For these families, the orbit of every point in the Julia set is either rigid, i.e. can be distinguished in some combinatorial terms from all other orbits, or it lands in the Julia set of a polynomial-like restriction of the original rational map. This is a statement of dynamical rigidity. It can also be phrased in a less dynamical way as a result on local connectivity of Julia sets along rigid orbits. In the talk, a guiding example for us will be rigidity of Newton maps which was recently established jointly with Dierk Schleicher. If time permits, we will also talk about related questions on rigidity in parameter space.