In this talk we will try to show the hidden geometry (symplectic and Poisson) in what we call b-manifolds. These manifolds were initially considered by Nest and Tsygan while studying formal deformations of symplectic manifolds with boundary and also by Melrose in the context of differential calculus and differential operators of manifolds with boundary. Symplectic b-manifolds lie between the symplectic and Poisson world. In particular, it is possible to prove local and semiglobal normal forms via b-de Rham theory. We will present these results and try to give constructive examples. This talk is based on joink work with Victor Guillemin and Ana Rita Pires.