  We develop a Thurston-like theory to characterize geometrically finite
rational maps, then apply it to study pinching and plumbing deformations of
rational maps. We show that in certain conditions the pinching path converges
uniformly and the quasiconformal conjugacy converges uniformly to a
semi-conjugacy from the original map to the limit. Conversely, every
geometrically finite rational map with parabolic points is the landing point of
a pinching path for any prescribed plumbing combinatorics.
