W. Thurston's theorems almost all aim to give a purely topological problem an appropriate geometry, or to identify an appropriate obstruction.. We will illustrate this in two examples:
--The Thurston pullback map to make a rational map from a post-critically finite branched cover of the sphere, and
--The skinning lemma, to find a hyperbolic structure for a Haken 3-manifold.
In both cases, either the relevant map on Teichmüller space has a fixed point, solving the geometrization problem, or there is an obstruction consisting a multicurve.[-]