Constant evolution of technology is leading to the improvement of electronical devices. Smaller, lighter, faster, are but a few of the properties that have been constantly improved, but these developments come hand in hand with negative downsides. In the case of miniaturization, this shortcoming is found in the inherent property of conducting materials-the limit of current density they can withstand before failure. This property, known as ampacity, is close to reaching its limits at the current scales of use, and the performances of some conductors such as gold or copper suffer severely from it. The need to find alternative conductors with higher ampacity is, therefore, an urgent need, but at the same time, one which requires simultaneous search for decreased density if it is to succeed in an ever-growing electronical world. The uses of these carbon nanotube-based materials, from airplane lightning strike protection systems to the microchip industry, will be evaluated, failure mechanisms at maximum current densities explained, limitations and difficulties in ampacity measurements with different size ranges evaluated, and future lines of research suggested. This review will therefore provide an in-depth view of the rare properties that make carbon nanotubes and their hybrids unique.