  In the early 1980's Thurston gave a topological characterization of rational
maps whose critical points have finite iterated orbits (\cite{Th,DH1}): given a
topological branched covering $F$ of the two sphere with finite critical
orbits, if $F$ has no Thurston obstructions then $F$ possesses an invariant
complex structure (up to isotopy), and is combinatorially equivalent to a
rational map.
  We extend this theory to the setting of rational maps with infinite critical
orbits, assuming a certain kind of hyperbolicity. Our study includes also
holomorphic dynamical systems that arise as coverings over disconnected Riemann
surfaces of finite type. The obstructions we encounter are similar to those of
Thurston. We give concrete criteria for verifying whether or not such
obstructions exist.
  Among many possible applications, these results can be used for example to
construct holomorphic maps with prescribed dynamical properties; or to give a
parameter description, both local and global, of bifurcations of complex
dynamical systems.
