Let $M^{2n}$ be a Poisson manifold with Poisson bivector field $Π$. We say that $M$ is b-Poisson if the map $Π^n:M\toΛ^{2n}(TM)$ intersects the zero section transversally on a codimension one submanifold $Z\subset M$. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of $(M,Π)$ 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.

34 pages. Some changes have been implemented mainly in Sections 2 and 6. Minor changes in exposition. References have been added