The notion of the *category of correspondences* of a category D with a specified, base change stable, class of morphisms S --- whose objects are those of D and whose morphisms are "spans" in D, one side of which belongs to S --- will be familiar to practitioners of Grothendieck's theory of motives. Perhaps less familiar is the fact that an obvious 2-categorical upgrade of correspondences has a universal property: it is the universal way to attach right adjoints to members of S subject to a base change formula. In the first talk, after providing some motivation and stating the main results, I will discuss notions of fibration and the Grothendieck fibration in higher category theory which provide a basic part of the framework for my arguments. Although they have their roots in Grothendieck's work on classical categories (SGA4, stack theory), these ideas take on a heightened importance in higher category theory, where they become absoutely indispensable.