  Based on \emph{ab initio} theoretical calculations of the optical spectra of
vertical heterostructures of MoSe$_2$ (or MoS$_2$) and WSe$_2$ sheets, we
reveal two spin-orbit-split Rydberg series of excitonic states below the
\textsl{A} excitons of MoSe$_2$ and WSe$_2$ with a significant binding energy
on the order of 250\,meV for the first excitons in the series. At the same
time, we predict crystalographically aligned MoSe$_2$/WSe$_2$ heterostructures
to exhibit an indirect fundamental band gap. Due to the type-II nature of the
MoSe$_2$/WSe$_2$ heterostructure, the indirect transition and the exciton
Rydberg series corresponding to a direct transition exhibit a distinct
interlayer nature with spatial charge separation of the coupled electrons and
holes. The experimentally observed long-lived states in photoluminescence
spectra of MoX$_2$/WY$_2$ heterostructure are attributed to such interlayer
exciton states. Our calculations further suggest an effect of stacking order on
the peak energy of the interlayer excitons and their oscillation strengths.
