Origami manifolds and b-symplectic manifolds are examples of manifolds which are symplectic except on a hypersurface Z: in the origami case, the symplectic form vanishes at Z, in the b-case, it explodes to infinity at Z. In this talk we will translate Delzant's classification of toric symplectic manifolds by their moment images to these two extreme cases, and see what that tells us about toric origami manifolds and toric b-symplectic manifolds.Ana Rita Pires,Cornell University