Author: Reinhard Diestel Title: A graph minor hierarchy. Abstract: The starting point for the Robertson-Seymour graph minor theory is the structural description of a particularly simple type of graph: of graphs that have {\it small path-widh\/}, ie which from far away look like fattened paths. Not all graphs have this structure, but among those that do not there are some minor-minimal ones: the (large binary) trees. The next step, then, is to describe the graphs that are (more complex than those of bounded path-width but) essentially no more complex than trees: graphs of small tree-width. Again, not all graphs have small tree-width, but among those that do not there are some minor-minimal ones: the grids. At this point, the graph minor theory takes a different turn - but why stop here? Isn't there a meaningful notion of grid-width, and can we not try to identify the ``minor-minimal graphs of unbounded grid-width''? And can we continue after that? This talk will outline a framework of how such a hierarchy of graph properties can indeed be defined. The aim is to create a structural yard-stick along which graph phenomena related to minors can be measured -- where in the hierarchy do they first occur, and where do they end -- and which may facilitate inductive proofs.