Because the thickness of Earth's surface plates is much smaller than their typical lateral dimensions, many aspects of their dynamics can be understood using mechanical `thin-sheet' theory. In this talk I use the `fluid' form of that theory together with numerical and experimental modeling to gain insight into three aspects of free (buoyancy-driven) subduction of oceanic lithosphere: (1) the mechanical interaction of a subducted slab with its surroundings in the upper mantle; (2) buckling instabilities of subducted lithosphere below the transition zone; and (3) the interaction of subducted slabs with the core-mantle boundary.