| $J$-holomorphic curves revolutionized the study of symplectic geometry when Gromov first introduced them in 1985. Through quantum cohomology, these curves are now linked to many of the most exciting new ideas in mathematical physics. This book presents the first coherent and full account of the theory of $J$-holomorphic curves, the details of which are presently scattered in various research papers. The first half of the book is an expository account of the field, explaining the main technical aspects. McDuff and Salamon give complete proofs of Gromov's compactness theorem for spheres and of the existence of the Gromov-Witten invariants. The second half of the book focuses on the definition of quantum cohomology. The authors establish that this multiplication exists, and give a new proof of the Ruan-Tian result that is associative on appropriate manifolds. They then describe the Givental-Kim calculation of the quantum cohomology of flag manifolds, leading to quantum Chern classes and Witten's calculation for Grassmannians, which relates to the Verlinde algebra. The Dubrovin connection, Gromov-Witten potential on quantum cohomology, and curve counting formulas are also discussed. The book closes with an outline of connections to Floer theory. |