Scientists and engineers can exploit nanostructures to manipulate thermal transport in solids. This is possible because the dominant heat carriers in nonmetals - crystal vibrations (or phonons) - have characteristic lengths in the nanometer range. We review research where this approach is used and propose future research directions. For instance, concepts such as phonon filtering, correlated scattering, and waveguiding could expand the extremes of thermal transport in both the insulating and conducting limits. This will have major implications on energy conservation and conversion, information technology, and thermal management systems.