The subject of thermal transport at the mesoscopic scale and in low-dimensional systems is interesting for both fundamental research and practical applications. As the first example of truly two-dimensional materials, graphene has exceptionally high thermal conductivity, and thus provides an ideal platform for the research. Here we review recent studies on thermal and thermoelectric properties of graphene, with an emphasis on experimental progresses. A general physical picture based on the Landauer transport formalism is introduced to understand underlying mechanisms. We show that the superior thermal conductivity of graphene is contributed not only by large ballistic thermal conductance but also by very long phonon mean free path (MFP). The long phonon MFP, explained by the low-dimensional nature and high sample purity of graphene, results in important isotope effects and size effects on thermal conduction. In terms of various scattering mechanisms in graphene, several approaches are suggested to control thermal conductivity. Among them, introducing rough boundaries and weakly-coupled interfaces are promising ways to suppress thermal conduction effectively. We also discuss the Seebeck effect of graphene. Graphene itself might not be a good thermoelectric material. However, the concepts developed by graphene research might be applied to improve thermoelectric performance of other materials.