Appears in collection : Geometry in non-positive curvature and Kähler groups

The interest of median geometry comes from its connections with property (T) and a-T-menability and, in its discrete version, with the solution to the virtual Haken conjecture. In this talk I shall explain how groups endowed with various forms of hyperbolic geometry, from lattices in rank one simple groups to acylindrically hyperbolic groups, present various degrees of compatibility with median geometry. This is on joint work with Indira Chatterji and joint work with John Mackay.