A metric space is multiended if it admits a bounded subset whose complement has at least two unbounded connected components. For instance, the line is multiended but higher-dimensional Euclidean spaces are not. In the late sixties, Stallings has given a remarkable characterization of those finitely generated groups whose Cayley graph is multiended; the only such torsion-free groups are free products of two nontrivial groups! A key part of the proof is the construction of a action on a tree; in the seventies, the study of general group actions on trees was achieved by Bass and Serre. In the meantime, the study of multiended Schreier graphs was started by Abels and Houghton, and a remarkable connection with nonpositively curved cube complexes was discovered by Sageev twenty years later. While outstanding applications of cube complexes have been made since then, I will try to focus on the question of understanding which finitely generated groups admit a multiended Schreier graph. Tea at 3:30 pm in SC 1425.