2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/147603We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are isolated then the space is a toric variety.71 pages, LaTeX. The paper is essentially rewritten (following an excellent referee report): I clarified statements, simplified proofs, added missing details, and corrected a false proof of a true statement, which is currently in Appendix CDifferential Geometry58F05, 70H33 (Primary) 53C12, 53C55 (Secondary)text