Most closed 4-manifolds do not admit symplectic forms, but most admit "near-symplectic forms", certain closed 2-forms which are symplectic outside of a collection of circles. This provides a gateway from the symplectic world to the non-symplectic world. Just like the Seiberg-Witten (SW) invariants, there are invariants in terms of J-holomorphic curves that are compatible with the near-symplectic form. Although the SW invariants don't apply to (potentially exotic) 4-spheres, nor do these spheres admit near-symplectic forms, there is still a way to bring in near-symplectic techniques.