An important aspect of Borel dynamics and graph combinatorics concerns the large-scale topological and geometric structure of Borel trees and graphs, as made precise by the notion of an end ("connected component at infinity") of a graph. For instance, by a result of Gaboriau and Ghys, a probability-measure-preserving graph with at least 3 ends per component must be non-amenable almost everywhere; the proof of this result uses a Borel version of Stallings' theorem on ends of groups. We will give a survey of some recent results in this area, that all share a theme of "doing Borel topology" on spaces of ends of graphs.