In this talk we generalize Hopf's famous classification of constant mean curvature spheres in R^3 to the general situation of classes of surfaces modeled by arbitrary elliptic PDEs in arbitrary three-manifolds, with the only hypothesis of the existence of a family of "candidate surfaces". In this way, we prove that any immersed sphere in such a class of surfaces is a candidate sphere. Among several applications, we solve two open problems of classical surface theory; we prove a 1956 conjecture by A.D. Alexandrov on the uniqueness of immersed spheres in R^3 that satisfy a general elliptic prescribed curvature equation, and show as a consequence that round spheres are the only elliptic Weingarten spheres immersed in R^3. Joint work with Jose A. Galvez.