This paper presents a brief review of experimental and theoretical studies on a three-dimensional heterostructure consisting of vertical carbon nanotubes (CNTs) connected perpendicularly to a graphene layer. This structure can serve as a potential building block for an all-carbon network in energy storage devices and on-chip interconnects. The review highlights reported works on the fabrication and characterization of such a heterostructure, with focus on the effect of the CNT-graphene interface on electrical conduction. While a direct comparison between experiment and theory is not possible at this time, a brief survey of theoretical efforts based on atomic cluster models nonetheless reveals important knowledge about the electronic transport properties of this all-carbon heterostructure.