Wave interference modifies phonon velocities and density of states, and in doing so creates forbidden energy bandgaps for thermal phonons. Materials that exhibit wave interference effects allow the flow of thermal energy to be manipulated by controlling the material's thermal conductivity or using heat mirrors to reflect thermal vibrations. The technological potential of these materials, such as enhanced thermoelectric energy conversion and improved thermal insulation, has fuelled the search for highly efficient phonon wave interference and thermal bandgap materials. In this Progress Article, we discuss recent developments in the understanding and manipulation of heat transport. We show that the rational design and fabrication of nanostructures provides unprecedented opportunities for creating wave-like behaviour of heat, leading to a fundamentally new approach for manipulating the transfer of thermal energy.