Van der Waals heterostructures are synthetic quantum materials composed of stacks of atomically thin two-dimensional (2D) layers. Because the electrons in the atomically thin 2D layers are exposed to layer-to-layer coupling, the properties of van der Waals heterostructures are defined not only by the constituent monolayers, but also by the interactions between the layers. Many fascinating electrical, optical and magnetic properties have recently been reported in different types of van der Waals heterostructures. In this Review, we focus on unique excited-state dynamics in transition metal dichalcogenide (TMDC) heterostructures. TMDC monolayers are the most widely studied 2D semiconductors, featuring prominent exciton states and accessibility to the valley degree of freedom. Many TMDC heterostructures are characterized by a staggered band alignment. This band alignment has profound effects on the evolution of the excited states in heterostructures, including ultrafast charge transfer between the layers, the formation of interlayer excitons, and the existence of long-lived spin and valley polarization in resident carriers. Here we review recent experimental and theoretical efforts to elucidate electron dynamics in TMDC heterostructures, extending from timescales of femtoseconds to microseconds, and comment on the relevance of these effects for potential applications in optoelectronic, valleytronic and spintronic devices.