*Abstract:* Gauge theory is rich in geometry. One particularly interesting notion is that of a spectral network on a punctured Riemann surface C, which can be motivated from the study of certain supersymmetric gauge theories. In this colloquium I will introduce spectral networks, and discuss a purely mathematical application: with a spectral network one can abelianize flat connections on the surface C. This will unify the description of shear as well as length-twist coordinates on Teichmueller space.