 In this review, we discuss a non-equilibrium thermodynamic theory for heat transport in superlattices, graded systems, and thermal metamaterials with defects. The aim is to provide researchers in non-equilibrium thermodynamics as well as material scientists with a framework to consider in a systematic way several non-equilibrium questions about current developments, which are fostering new aims in heat transport, and the techniques for achieving them. For instance, defect engineering, dislocation engineering, stress engineering, phonon engineering, and nanoengineering. We also suggest some new applications in the particular case of mobile defects. This article was authored by David Jue and Liliana Rastucia.