A near-symplectic manifold is a four-manifold equipped with a two-form that is symplectic on the complement of a union of circles and that vanishes ˇ§nicelyˇ¨ along the circles. In this talk I will discuss what closed four-manifolds admit a near-symplectic structure that is invariant under a torus action, or more generally, compatible with a Lagrangian fibration with toric singularities. In particular, toric near-symplectic manifolds are classified by generalized moment map images. The study of such structures on near-symplectic manifolds is motivated by Taubesˇ¦ program to develop Gromov-Witten type invariants for near-symplectic manifolds and recent calculations of Gromov-Witten invariants of toric manifolds in terms of graphs in moment map images (due to Parker using symplectic field theory, and Mikhalkin using tropical algebraic geometry). This is joint work with David Gay.