In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of the main theorem of Hamilton in [19]; the other is to extend some results of Perelman [27], [28] to four-manifolds. During the proof we have actually provided, parallel to the paper of the second author with H.-D. Cao [3], all necessary details for the part from Section 1 to Section 5 of Perelman’s second paper [28] on the Ricci flow.