Sheaves have long been classical tools for studying the topology of manifolds. Symplectic geometry, which encodes topological information about a manifold via its cotangent bundle, has revealed a profound connection to sheaf theory through the microlocal framework developed by Kashiwara and Schapira. Remarkably, many important symplectic invariants can now be computed using sheaves. In this talk, I will survey several well-known applications of sheaf theory in symplectic geometry and also consider the reverse perspective: how symplectic geometry provides constructions and insights that deepen our understanding of sheaf theory. This latter viewpoint is central to obtaining a global version of the microlocal Riemann-Hilbert correspondence in joint work with Côté, Nadler, and Shende.