  We complete the classification of Hamiltonian torus and circle actions on
symplectic four-dimensional manifolds. Following work of Delzant and Karshon,
Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic
four-manifold were characterized by Karshon, Kessler and Pinsonnault. What
remains is to study the case of Hamiltonian actions on blowups of S^2-bundles
over a Riemann surface of positive genus. These do not admit 2-torus actions.
In this paper, we characterize Hamiltonian circle actions on them. We then
derive combinatorial results on the existence and counting of these actions. As
a by-product, we provide an algorithm that determines the g-reduced form of a
blowup form. Our work is a combination of "soft" equivariant and combinatorial
techniques, using the momentum map and related data, with "hard" holomorphic
techniques, including Gromov-Witten invariants.
