This volume introduces Riemann's approach to multiple-value functions and the geometrical representation of these functions by Riemann surfaces. It concentrates on the type of functions that can be defined upon these surfaces, restricting the treatment to rational functions and their integrals demonstrating how Riemann's mathematical ideas about Abelian integrals can be derived by reflecting upon the flow of electric current on surfaces. This work is one of the best introductions to the origins of topological problems.