We will review the progress made in the past decade in thermal transport measurements of one-dimensional(1D)and two-dimensional(2D)materials.The reported temperature-and sizedependent thermal conductivity data of carbon nanotubes,graphene and other 1D and 2D systems will be examined in conjunction with the experimental techniques employed to obtain these data.The objective is to identify gaps that may still exist between the available experimental data and the theoretical predictions of the unique phonon transport behaviors in 1D and 2D systems,and to suggest possible future directions.