Abstract: The purpose of this talk is to explore the relationship between a manifold and its totally geodesic submanifolds. The two primary directions we will explore are motivated by the following pair of questions. First, how much global geometry is determined by a manifold's submanifolds? Second, what relationship exists between the (co)homology of the manifold and the (co)homology of its submanifolds? In the talk, we restrict ourselves to arithmetic locally symmetric manifolds. Aside from being a beautiful, natural, explicit family of examples, the geometry/cohomology has important and direct connections to many other areas of mathematics. Various bits of this material are separately joint with Benson Farb, Amir Mohammadi, and Alan Reid.