© 2018 The Royal Society of Chemistry. Since its discovery, graphene has been a promising material for spintronics: its low spin-orbit coupling, negligible hyperfine interaction, and high electron mobility are obvious advantages for transporting spin information over long distances. However, such outstanding transport properties also limit the capability to engineer active spintronics, where strong spin-orbit coupling is crucial for creating and manipulating spin currents. To this end, transition metal dichalcogenides, which have larger spin-orbit coupling and good interface matching, appear to be highly complementary materials for enhancing the spin-dependent features of graphene while maintaining its superior charge transport properties. In this review, we present the theoretical framework and the experiments performed to detect and characterize the spin-orbit coupling and spin currents in graphene/transition metal dichalcogenide heterostructures. Specifically, we will concentrate on recent measurements of Hanle precession, weak antilocalization and the spin Hall effect, and provide a comprehensive theoretical description of the interconnection between these phenomena.