  In this paper we develop an abstract theory for the Codazzi equation on
surfaces, and use it as an analytic tool to derive new global results for
surfaces in the space forms ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. We give
essentially sharp generalizations of some classical theorems of surface theory
that mainly depend on the Codazzi equation, and we apply them to the study of
Weingarten surfaces in space forms. In particular, we study existence of
holomorphic quadratic differentials, uniqueness of immersed spheres in
geometric problems, height estimates, and the geometry and uniqueness of
complete or properly embedded Weingarten surfaces.
