Abstract: Given an almost complex manifold M (with a prescribed complex vector-bundle structure J on TM the J-holomorphic cureves are maps from the Riemann sphere CP¹ into M whose derivatives are complex linear and hence they satisfy a nonlinear Cauchy-Riemann equation. Under suitable assumptions the modulii space of such maps is a manifold. The tangent space is described by a family of linear operators(now parametrized by the space of solution itself). Such moduli spaces provide very useful invariants for symplectic manifolds. The talk shall present the standard notions of J-holomorphic curves with slightly operator theoretic inclination.