The microlocal theory of sheaves, developed by Kashiwara and Schapira, reveals a connection between sheaves on a manifold M and Lagrangian subsets of the cotangent bundle of M. The theory includes a package of "microlocal sheaf operations" that generalize, for instance, the operation of restricting a sheaf to an open subset. In the 1980s Kazhdan and Laumon proposed gluing categories of sheaves along these microlocal operations, for applications in representation theory. The work of Nadler and Zaslow and some recent ideas of Kontsevich suggest that the Fukaya categories of certain symplectic manifolds can be modeled in Kazhdan-Laumon style--a model that uses a minimum of symplectic geometry. I will explain some of these constructions and their relevance to Kontsevich's homological mirror symmetry conjecture, which predicts an equivalence between Fukaya categories and derived categories of coherent sheaves. The parts of this talk that are original are based on joint work with Nicol\'o Sibilla and Eric Zaslow.