The optical spectra of vertically stacked MoSe2/WSe2 heterostructures contain additional "interlayer" excitonic peaks that are absent in the individual monolayer materials and exhibit a significant spatial charge separation in out-of-plane direction. A many-body perturbation theory approach is used to simulate the excitonic spectra of MoSe2/WSe2 heterobilayers with three stacking orders, considering both momentum-direct and momentum-indirect excitons. The small oscillator strengths and the optical responses of the interlayer excitons are significantly stacking-dependent and give rise to high radiative lifetimes in the range of 5-200 ns at low temperature for the "bright" interlayer excitons. Solving the finite-momentum Bethe-Salpeter Equation (BSE), the lowest energy excitation is predicted to be an exciton over the fundamental indirect band gap, with a binding energy of 220 meV. However, in agreement with recent magneto-optics experiments and previous theoretical studies, the simulations of the effective excitonic g-factors suggest that the low energy momentum-indirect excitons are not experimentally observed. The existence of "interlayer" C excitons with significant exciton binding energies and optical oscillator strengths is further revealed, which are analogous to the prominent band nesting excitons in mono- and few-layer transition-metal dichalcogenides.