Median spaces form a broad and increasingly important class of metric spaces, encompassing both CAT(0) cube complexes and real trees. The study of finitely generated groups admitting free transitive (or proper cocompact) actions on discrete median spaces—equivalently, on the 0-skeletons of CAT(0) cube complexes—are an active area of study. In contrast, much less is understood about their continuous analogue: groups acting freely and transitively on connected median spaces. I will present several methods for constructing such actions, focusing on actions on real trees and their products, and discuss some of the surprising behaviours that show up. Even when considering real trees, the class of groups acting on such spaces is vastly more diverse than in the discrete setting: while any simplicial tree admits at most one free vertex transitive action, we will see that there are 2^{2^{\aleph_0}} pairwise non-isomorphic groups which admit a free transitive action on the universal real tree with continuous valence.