I review recent progress in mathematical physics that aims at generalizing familiar concepts from differential geometry in a way that is adapted to the structure of the Einstein equations in String and M-theory. In this context, a notion of generalized Lie derivative is introduced that is covariant w.r.t. certain Lie groups that appear as "duality groups" in string theory. These structures are defined on an extended space/generalized spacetime, but the functions on that space are restricted by a "section constraint" that effectively reduces the space to the physical spacetime of string theory. The generalized Lie derivatives satisfy an algebra that is governed by a generalized Lie bracket, which is a generalization of the Courant bracket in generalized geometry and thus has a non-trivial Jacobiator. I discuss the consequences of these novel structures for the definition of connections and curvatures. Finally, I discuss recent work on a generalization of Yang-Mills theory, with gauge connections based on a Courant-type bracket rather than a Lie bracket.