This talk will mainly be about applications of complex function theory to inverse spectral problems for canonical systems, which constitute a broad class of second order differential equations. I will start with the basics of Krein-de Branges theory, then present an algorithm for inverse spectral problems developed by Makarov and Poltoratski for locally-finite periodic spectral measures. I will then extend the algorithm to certain classes of non-periodic spectral measures and present several examples. If time permits, I will talk about the connection between inverse spectral problems and nonlinear Fourier transforms. This is joint work with Alexei Poltoratski.