  We characterize the current-carrying capacity (CCC), or ampacity, of
highly-conductive, light, and strong carbon nanotube (CNT) fibers by measuring
their failure current density (FCD) and continuous current rating (CCR) values.
We show, both experimentally and theoretically, that the CCC of these fibers is
determined by the balance between current-induced Joule heating and heat
exchange with the surroundings. The measured FCD values of the fibers range
from 10$^7$ to 10$^9$ A/m$^2$ and are generally higher than the previously
reported values for aligned buckypapers, carbon fibers, and CNT fibers. To our
knowledge, this is the first time the CCR for a CNT fiber has been reported. We
demonstrate that the specific CCC (i.e., normalized by the linear mass density)
of our CNT fibers are higher than those of copper.
