Gromov convergence of pseudoholomorphic disks

The goal of this paper is to give a self-contained exposition of Gromov compactness for pseudoholomorphic disks in compact symplectic manifolds. The proof leads naturally to the concept of stable maps which was first introduced by M. Kontsevich. Our definition of stable maps for disks is based on the one given by D. McDuff and D. Salamon for spheres. We also generalize the notion of Gromov convergence to the case of disks. We show that the homotopy class is preserved under Gromov convergence. Dedicated to Vladimir Arnold