Atomically thin transition-metal dichalcogenides (TMDs) exhibit excellent optoelectronic properties and are promising for fundamental investigations and applications in next-generation photovoltaics, photodetectors, quantum sensors, and quantum computers. Combined with other semiconductor materials, TMDs can form heterostructures with improved and even emergent properties. The type of interface between constituent components of the heterostructure ultimately defines the heterostructure's behavior and performance. Understanding charge transfer at the TMD/semiconductor interface is vital to the development of next-generation photovoltaic, photodetector, and quantum materials applications. Here we review recent studies of photoinduced charge transfer in TMD/semiconductor heterostructures, where the semiconductor constituent is a TMD, a colloidal quantum dot (QD), or a hybrid perovskite. We focus on experimental studies that utilize ultrafast optical methods to decipher the charge-transfer kinetics. We present examples of the formation of interlayer excitons as a proof of charge transfer, interlayer excitons trapped in moire potentials, and heterostructures showing spin-valley conservation of interlayer exciton emission. We review TMD/ QD heterostructures where charge transfer is tuned by energy-band-gap matching (engineering) and close the review with a subfield on TMD/perovskite heterostructures.