  Van der Waals heterostructures of two-dimensional transition metal
dichalcogenides provide a unique platform to engineer optoelectronic devices
tuning their optical properties via stacking, twisting, or straining. Using ab
initio Many-Body Perturbation Theory, we predict the electronic and optical
(absorption and photoluminescence spectra) properties of MoS$_2$/WS$_2$ and
MoSe$_2$/WSe$_2$ hetero-bilayers with different stacking and twisting. We
analyse the valley splitting and optical transitions, and explain the
enhancement or quenching of the inter- and intra-layer exciton states. Contrary
to established models, that focus on transitions near the high-symmetry point
K, our results include all possible transitions across the Brillouin Zone. This
result, for a twisted Se-based heterostructures, in an interlayer exciton with
significant electron density in both layers and a mixed intralayer exciton
distributed over both MoSe$_2$ and WSe$_2$. We propose that it should be
possible to produce an inverted order of the excitonic states in some
MoSe$_2$/WSe$_2$ heterostructures, where the energy of the intralayer WSe$_2$
exciton is lower than that in MoSe$_2$. We predict the variability of the
exciton peak positions ($\sim$100 meV) and the exciton radiative lifetimes,
from pico- to nano-seconds, and even micro-seconds in twisted bilayers. The
control of exciton energies and lifetimes paves the way towards applications in
quantum information technologies and optical sensing.
