Exciton binding energies of hundreds of meV and strong light absorption in the optical frequency range make transition metal dichalcogenides (TMDs) promising for novel optoelectronic nanodevices. In particular, atomically thin TMDs can be stacked to heterostructures enabling the design of new materials with tailored properties. The strong Coulomb interaction gives rise to interlayer excitons, where electrons and holes are spatially separated in different layers. In this work, we reveal the microscopic processes behind the formation, thermalization and decay of these fundamentally interesting and technologically relevant interlayer excitonic states. In particular, we present for the exemplary MoSe2-WSe2 heterostructure the interlayer exciton binding energies and wave functions as well as their time- and energy-resolved dynamics. Finally, we predict the dominant contribution of interlayer excitons to the photoluminescence of these materials.