In their seminal study of four-dimensional supersymmetric QFTs "of class S" early last decade, Gaiotto, Moore, and Neitzke proposed a rich conjectural picture relating Higgs bundles, Donaldson-Thomas theory, hyperkähler geometry, WKB analysis, and more, via a fundamental object called a spectral network.  This lecture series aims to give an introduction to some of their ideas and the progress that has been made since. In the first lecture, I'll describe the basic physical setup and explain some motivations regarding counting of so-called BPS states. In the second lecture, I'll define precisely spectral networks for quadratic as well as cubic differentials (with some indications of the general case), and explain what is and isn't known about them. In the third lecture, I'll describe some applications to the geometry of spaces of stability conditions, and discuss the Riemann-Hilbert problem which can be used to obtain so-called Joyce structures on them.