The material hybridization and multi-scale integration is considered to be an important direction of research in modern electronics. Simultaneous use of graphene and organic semiconductors in reduced-dimensional device architectures is a promising approach in this context, as they possess both structural compatibility and functional complementariness. In this presentation, we overview recent advances in understanding and exploitation of physical properties of the characteristic van der Waals interface formed between these materials. We first describe fundamental issues in charge transport and injection associated with the traps in fully depleted organic metal-contact diodes, which form the foundation of the advanced graphene/organic junction devices. We then explore the unique templating property of graphene as a bottom injection contact for the growth of vacuum-evaporated molecular semiconductors, which leads us to demonstrate the efficient single-layer organic rectifiers. The results from broad-band impedance spectroscopy on these devices are discussed in detail. Finally, we introduce the high-performance low-voltage graphene/organic heterojunction field-effect transistors, realized from the vertical integration of a junction diode and external gating. This new generation of devices shows a particularly promising switching performance with high current densities, and we focus on the mechanism behind the asymmetrical electrical transport and substantial work-function modulation at the graphene/organic interface through inverse modeling approaches, and suggest a strategy for further performance improvement.