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.