A symplectic form on a 4-manifold gives us a reasonable
grip on its topology, but often no such form exists. The idea of near-symplectic geometry is to relax the definition just enough to prove an existence theorem, but not so much as to lose the geometry. We don't know how to do this in higher dimensions. I'll explain what near-symplectic structures are, why they exist, and what they might be good for. Symplectic experience will not be required.