We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory, and at the same time serves as a rich source of new invariants of contact manifolds and their Legendrian submanifolds. Moreover, we hope that the applications of SFT go far beyond this framework.

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [4]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov's compactness theorem in [8] as well as compactness theorems in Floer homology theory, [6,7], and in contact geometry, [9,19].

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