A long time ago Leray and Oka introduced sheaf theory as a format for studying functions (smooth, holomorphic, etc.) on a manifold according to their local properties and constraints. Over the 1980s, with applications to PDEs in mind, Kashiwara and Schapira developed a so-called microlocal version of sheaf theory. Microlocal sheaf theory supplements the local measurements of sheaves at co-ordinates with microlocal measurements in co-directions. Though it has analytic origins the subject is very soft and algebraic. Many of its applications have been in representation theory.Recently, the Kashiwara-Schapira theory has had striking interactions with symplectic geometry. An attention-catching feature of these interactions is that they avoid Gromov's theory of J-holomorphic disks and anyhard analysis, which for a long time has been the most powerful symplectic tool. In some sense microlocal sheaf theory lets you buy some of the same kinds of theorems with Arnold's theory of wavefront singularities, instead of with J-holomorphic disks. I will discuss some of these developments, present an extension of the theory to sheaves of spectra that is joint with Xin Jin, and a sheaf-theoretic approach to computing open Gromov-Witten invariants that is joint with Eric Zaslow.