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 MoS2/WS2 and MoSe2/WSe2 heterobilayers with different stacking and twisting. We analyze the valley splitting and optical transitions, and we explain the enhancement or quenching of the interand intralayer exciton states. We fully include transitions within the entire Brillouin Zone, contrary to predictions based on continuum models which only consider energies near the K point. As a result, we predict an interlayer exciton with significant electron density in both layers and a mixed intralayer exciton distributed over both MoSe2 and WSe2 in a twisted Se-based heterostructure. We propose that it should be possible to produce an inverted order of the excitonic states in some MoSe2/WSe2 heterostructures, where the energy of the intralayer WSe2 exciton is lower than that in MoSe2. We predict the variability across different stacking of the exciton peak positions (-100 meV) and the exciton radiative lifetimes, from pico- to nanoseconds, and even microseconds in twisted bilayers. The control of exciton energies and lifetimes paves the way toward applications in quantum information technologies and optical sensing.