This review discusses recent advances in materials engineering to control thermal conductivity. We begin by presenting theories of heat conduction for general material classes, focusing on common approximations and trends. Next, we discuss characterization techniques for measuring thermal conductivity and the underlying transport properties. Advanced materials at the frontiers of thermal transport, such as rattlers, complex unit cells, nanowires, and nanocomposites, are treated in depth using experimental data and theoretical predictions. The review closes by highlighting several promising areas for further development.