  van der Waals heterostructures consisting of vertically stacked
transition-metal dichalcogenides (TMDs) exhibit a rich landscape of bright and
dark intra- and interlayer excitons. In spite of a growing literature in this
field of research, the type of excitons dominating optical spectra in different
van der Waals heterostructures has not yet been well established. The spectral
position of exciton states depends strongly on the strength of hybridization
and energy renormalization due to the periodic moir\'e potential. Combining
exciton density-matrix formalism and density-functional theory, we shed light
on the exciton landscape in TMD homo- and heterobilayers at different
stackings. This allows us to identify on a microscopic footing the
energetically lowest-lying exciton state for each material and stacking.
Furthermore, we disentangle the contribution of hybridization and layer
polarization-induced alignment shifts of dark and bright excitons in
photoluminescence spectra. By revealing the exciton landscape in van der Waals
heterostructures, our work provides the basis for further studies of the
optical, dynamical, and transport properties of this technologically promising
class of nanomaterials.
