Let M2n be a Poisson manifold with Poisson bivector field . We say thatM is b-Poisson if the map n : M ! 2n(TM) intersects the zero section transversally on a codimension one submanifold Z M. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of (M, ) in the neighbourhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology