Van der Waals heterostructures (vdWHs) of vertically stacked two-dimensional (2D) atomic crystals have been used to elicit intriguing phenomena stemming from strong electronic correlations, magnetic textures, and interlayer excitons spawned at the heterointerface. However, vdWHs comprised of heterointerfaces between these 2D atomic crystal lattices and molecular assemblies are emerging as equally intriguing platforms supporting properties to be harnessed for photovoltaic energy conversion, photodetection, spin-selective charge injection, and quantum emission. In this perspective, we summarize recent research examining exciton dynamics in heterostructures between semiconducting 2D transition metal dichalcogenides and molecular organic semiconductors. We discuss methods for assembly of these heterostructures, the nature of interlayer or charge-transfer excitons at transition-metal dichalcogenide (TMD)-molecule interfaces, explicit exciton transfer between organics and TMDs, and other interfacial phenomena driven by the merger of these two material classes. We also suggest key new research directions extending the remit of these 2D atomic-molecular lattice heterointerfaces into the domains of condensed matter physics, quantum sensing, and energy conversion.