Structure of group-labeled graphs

Kuratowski's theorem states that a graph is embeddable on the plane if and only if it does not contain a subgraph homeomorphic to K_5 or K_{3,3}. This was vastly generalized by Robertson and Seymour in their Graph Minors Project in which surface embeddings play a central role in proving that graphs are well-quasi-ordered by the minor relation. I will describe some of the highlights of this theory and discuss some applications to my work on group-labeled graphs.