This paper is concerned with Floer cohomology of manifolds with contact type boundary. In this case, there is no conjecture on this ring, as opposed to the compact case, where it is isomorphic to the usual cohomology (with the quantum product). We construct two mappings in Floer cohomology and prove some functorial properties of these two mappings. The first one is a map from the Floer cohomology of M to the relative cohomology of M modulo its boundary. The other is associated to a codimension zero embedding, and may be considered as a cohomological transfer. These maps are used to define some properties of symplectic manifolds with contact type boundary. These are algebraic versions of the Weinstein conjecture, asserting existence of closed characteristics on ∂M. This is proved for many cases, Euclidean space and subcritical Stein manifolds, vector bundles, products, cotangent bundles. It is also proved that the above property implies some restrictions on Lagrangian embeddings, and also yields in certain cases, existence results for holomorphic curves bounded by the Lagrange submanifold. The last section is devoted to applications of this existence result, to real forms of Stein manifolds and obstructions to polynomial convexity in Stein manifolds. Many of our applications rely on the computation of the Floer cohomology of a cotangent bundle, that is the subject of Part II.

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