In this thesis we study a number of geometric structures arising in the study of four-dimensional supersymmetric quantum field theories. We study properties and applications of so-called "spectral networks" on Riemann surfaces C, focusing in particular on the "abelianization map" which we use to produce special coordinate systems on moduli spaces of local systems on C. We generalize the classical Fenchel-Nielsen coordinates and utilize these coordinates to compute superpotentials, following and generalizing a conjecture of Nekrasov-RoslyShatashvili. Our first result is a computation of the higher rank spectral coordinates associated to certain "generalized Fenchel-Nielsen" networks, yielding explicit formulas for the trace functions on the moduli space with two "minimal" and two "maximal" punctures. We use this result to verify the NRS conjecture at the lowest order asymptotics for a prototypical SU(3) theory, and furthermore compute the 1-instanton correction in the SU(2) case, extending previous results. In the final chapter we include some partial results the author has obtained on the existence and uniqueness of abelianizations for certain classes of networks related to Grassmannians.