  We study rigidity of rational maps that come from Newton's root finding
method for polynomials of arbitrary degrees. We establish dynamical rigidity of
these maps: each point in the Julia set of a Newton map is either rigid (i.e.
its orbit can be distinguished in combinatorial terms from all other orbits),
or the orbit of this point eventually lands in the filled-in Julia set of a
polynomial-like restriction of the original map. As a corollary, we show that
the Julia sets of Newton maps in many non-trivial cases are locally connected;
in particular, every cubic Newton map without Siegel points has locally
connected Julia set. In the parameter space of Newton maps of arbitrary degree
we obtain the following rigidity result: any two combinatorially equivalent
Newton maps are quasiconformally conjugate in a neighborhood of their Julia
sets provided that they either non-renormalizable, or they are both
renormalizable ``in the same way''. Our main tool is the concept of complex box
mappings due to Kozlovski, Shen, van Strien; we also extend a dynamical
rigidity result for such mappings so as to include irrationally indifferent or
renormalizable situations.
