Heat flow in nanomaterials is an important area of study, with both fundamental and technological implications. However, little is known about heat flow in two-dimensional devices or interconnects with dimensions comparable to the phonon mean free path. Here we find that short, quarter-micron graphene samples reach ~35% of the ballistic thermal conductance limit up to room temperature, enabled by the relatively large phonon mean free path (~100 nm) in substrate-supported graphene. In contrast, patterning similar samples into nanoribbons leads to a diffusive heat-flow regime that is controlled by ribbon width and edge disorder. In the edge-controlled regime, the graphene nanoribbon thermal conductivity scales with width approximately as ~W(1.8)(0.3), being about 100 W m(-1) K(-1) in 65-nm-wide graphene nanoribbons, at room temperature. These results show how manipulation of two-dimensional device dimensions and edges can be used to achieve full control of their heat-carrying properties, approaching fundamentally limited upper or lower bounds.