I will describe a coherent-constructible correspondence, which is a monoidal dg functor between the category of equivariant perfect sheaves of a toric variety and the category of some constructible sheaves over a real vector space (dual Lie algebra of the torus). This correspondence is motivated by homological mirror symmetry - performing a T-duality on an equivariant line bundle produces a Lagrangian submanifold in the cotangent bundle of that dual Lie algebra. The Nadler-Zaslow's theorem then equates this Lagrangian to a constructible sheaf on the base. This correspondence can be extended to toric DM stacks, and as an exercise, one may show the derived categories are equivalent from the combinatorial constructible viewpoint in the case of some toric crepant resolutions. (This talk is based on the joint works with Chiu-Chu Liu, David Treumann and Eric Zaslow)