[en] We have produced nanotube-in-oil suspensions and measured their effective thermal conductivity. The measured thermal conductivity is anomalously greater than theoretical predictions and is nonlinear with nanotube loadings. The anomalous phenomena show the fundamental limits of conventional heat conduction models for solid/liquid suspensions. We have suggested physical concepts for understanding the anomalous thermal behavior of nanotube suspensions. In comparison with other nanostructured materials dispersed in fluids, the nanotubes provide the highest thermal conductivity enhancement, opening the door to a wide range of nanotube applications