Optimal branched transport is a variant of the Monge-Kantorovich problem in which transportation networks exhibit tree-like structures, reflecting the efficiency of grouping mass along shared paths. This talk introduces the problem and reviews results on its well-posedness, using tools from geometric measure theory. Special attention will be given to the mailing problem, where source-target pairs are prescribed. I will discuss a recent extension of the fundamental stability and regularity results to all transportation regimes. Based on joint work in progress with A. De Rosa and J. Krukowski.