  An origami manifold is a manifold equipped with a closed 2-form which is
symplectic except on a hypersurface where it is like the pullback of a
symplectic form by a folding map and its kernel fibrates with oriented circle
fibers over a compact base. We can move back and forth between origami and
symplectic manifolds using cutting (unfolding) and radial blow-up (folding),
modulo compatibility conditions. We prove an origami convexity theorem for
hamiltonian torus actions, classify toric origami manifolds by polyhedral
objects resembling paper origami and discuss examples. We also prove a
cobordism result and compute the cohomology of a special class of origami
manifolds.
