The recent intensive interaction between mathematics and theoretical physics has brought to mathematics a number of new important structures. An excellent example of this phenomenon is the mirror symmetry conjecture and, in particular, the Gromow-Witten invariants and quantum cohomology. The monograph concentrates on the theory of J-holomorphic curves, its role in symplectic topology and its relations to new structures coming from physics. In the monograph, the reader can find a careful description of the foundation of the theory of J-holomorphic curves. The first part of the book describes the Fredholm theory and compactness results for J-holomorphic spheres and discs. The second part of the book starts with the definition of the Gromow-Witten invariants. Locally Hamiltonian fibrations over Riemann surfaces are used in the definition of the Gromow-Witten invariants for arbitrary genus and for various applications of the theory. The gluing theorem is proved and used for a study of quantum cohomology, the Gromow-Witten potential and Frobenius manifolds. Some prerequisities (properties of linear elliptic operators, the Fredholm theory and implicit function theorem for Banach manifolds, the Riemann-Roch theorem for manifolds with boundary, the moduli space of stable curves of zero genus and positivity of intersections and the adjunction inequality for J-holomorphic curves in dimension four) are collected in appendices. The book offers a systematic treatment of one of important new fields in mathematics and should be available in every library.

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