I will outline efforts to describe the Kasparov product of two unbounded Kasparov modules in terms of another unbounded module. The reasons for doing this are overwhelmingly computational, but also to help us delineate what is geometric in noncommutative geometry. Along with a history/summary of what has been done, I will present (very briefly) two types of application: the bulk-edge correspondence and noncommutative dynamical systems. This talk is based on the work of numerous people, but strongly influenced by my joint work with Bram Mesland.