This thesis presents an experimental study of phonon transport in individual suspended single-wall carbon nanotubes (SWCNTs). A microfabricated device consisting of two adjacent suspended membranes, each with a platinum resistance heater and thermometer, was used to directly measure the thermal conductance of three individual SWCNTs. These results show the effects of Umklapp phonon-phonon scattering remain weak and the thermal conductance remains roughly proportional to the ballistic conductance throughout the temperature range of 100 to 490 K. Additionally, through the use of transmission electron microscopy analysis we have for the first time directly obtained the diameter of a nanotube for which thermal measurements were obtained and determined the thermal conductivity of this SWCNT. The thermal conductivity of this 1.6 nm diameter, 4.72 μm long nanotube increases with temperature as ~T[superscript 1.5] throughout the temperature range indicating static scattering processes dominate transport in this regime. The measured thermal conductivity is greater than 1000 W/m·K above room temperature making it one of the best thermal conductors known.