  We prove a dynamical Shafarevich theorem on the finiteness of the set of
isomorphism classes of rational maps with fixed degeneracies. More precisely,
fix an integer d at least 2 and let K be either a number field or the function
field of a curve X over a field k, where k is of characteristic zero or p>2d-2
that is either algebraically closed or finite. Let S be a finite set of places
of K. We prove the finiteness of the set of isomorphism classes of rational
maps over K with a natural kind of good reduction outside of S. We also prove
auxiliary results on finiteness of reduced effective divisors in
$\mathbb{P}^1_K$ with good reduction outside of S and on the existence of
global models for rational maps.
