The uniformization theorem of Riemann surfaces is one of the most beautiful and important theorems in mathematics. Besides giving a clean classification of Riemann surfaces, its proof has motivated many new methods, such as the Riemann–Hilbert correspondence, Picard–Fuchs equations, and higher-dimensional generalizations of the uniformization theorem, which include Calabi–Yau manifolds.This volume consists of expository papers on the four topics in its title, written by experts from around the world, and is the first to put forth a comprehensive discussion of these topics, and of the relations between them. As such, it is valuable as an introduction for beginners, and as a reference for mathematicians in general.