Spectral networks are certain decorated graphs drawn on a Riemann surface. I will describe a conjectural picture in which spectral networks can be viewed as analogues of Hermitian-Einstein metrics on vector bundles, and in which holomorphic differentials on the Riemann surface arise as stability conditions on certain Fukaya-type categories. This talk is based on various joint projects with Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich, and Carlos Simpson.