A basic problem in differentiable dynamical systems is to understand how a robust dynamical property (meaning a property that holds for a system as well as all nearby systems) on the underlying manifold would influence the behavior of the tangent map on the tangent bundle. In this talk we discuss some recent and some ongoing works on the robust dynamics of C^1 generic diffeomorphisms. No prior knowledge of smooth dynamics will be assumed; all necessary background and definitions will be provided during the talk.