Bipartite graph embeddings, Riemann surfaces and Galois groups

This is a survey of two related research topics, in each of which surface embeddings of bipartite graphs provide a bridge between two separate areas of mathematics: in the first case these are Riemann surface theory and the Galois theory of algebraic number fields; in the second case they are topological graph theory and finite group theory. We give explicit examples to illustrate the connection in each case.