Explore the fourth lecture in a series on median spaces and algebras, delivered by Elia Fioravanti as part of the Workshop on Cube Complexes and Combinatorial Geometry. Delve into the fascinating world of median spaces, which generalize L^1 spaces, CAT(0) cube complexes, and R-trees. Discover how these structures play crucial roles in geometric group theory, appearing in both finite and infinite-dimensional contexts. Examine their applications in studying quasi-flats, quasi-isometries, and automorphisms for various groups, including those with HHS structures and cocompactly cubulated groups. Investigate the connections between infinite-dimensional median spaces and functional analytic properties of groups, such as Kazhdan's property (T) and the Haagerup property. Gain insights into the theory of median algebras and its implications for cocompactly cubulated groups in this 28-minute presentation from the Centre de recherches mathématiques.