Power consumption and heat dissipation are significant challenges in electronics ranging from mobile devices to large data centers. A fundamental examination of energy dissipation in such contexts can lead to orders of magnitude improvements in energy efficiency. We present recent highlights from our work examining power and heat dissipation in nanoscale device geometries, at contacts or interfaces, and when novel materials are involved. For instance, thermal conductivity is significantly reduced in nanostructures due to the role of boundary scattering. However relatively unusual effects such as quasi-ballistic and thermoelectric transport could be used to partially mitigate the heat generated during device operation. In addition, careful low-power device design from the outset can alleviate heat dissipation problems before they begin. For example, data storage based on phase-change (rather than charge) with carbon nanotube electrodes can lead to two orders of magnitude reduction in power dissipation. The results suggest much room to improve power dissipation in nanoscale electronics, towards near-fundamental limits, through the co-design of geometry and materials.