The extremely high thermal conductivities of carbon nanotubes have motivated a wealth of research. Progress includes innovative conduction metrology based on microfabricated platforms and scanning thermal probes, as well as simulations exploring phonon dispersion and scattering using both transport theory and molecular dynamics. This article highlights these advancements as part of a detailed review of heat conduction research both on individual carbon nanotubes and nanostructured films consisting of arrays of nanotubes or disordered nanotube mats. Nanotube length, diameter, and chirality strongly influence the thermal conductivities of individual nanotubes and the transition from primarily diffusive to ballistic heat transport with decreasing temperature. A key experimental challenge, for both individual nanotubes and aligned films, is the separation of intrinsic and contact resistances. Molecular dynamics simulations have studied the impacts of specific types of imperfections on the nanotube conductance and its variation with length and chirality. While the properties of aligned films fall short of predictions based on individual nanotube data, improvements in surface engagement and post-fabrication nanotube quality are promising for a variety of applications including mechanically compliant thermal contacts.