Coulomb bound electron-hole pairs, excitons, govern the optical properties of semi-conducting transition metal dichalcogenides like MoS$_2$ and WSe$_2$. We study optical transitions at the K-point for 2H homobilayer MoS$_2$ in Density Functional Theory (DFT) including excitonic effects and compare with reflectivity measurements in high quality samples encapsulated in hexagonal BN. In both calculated and measured spectra we find a strong interlayer exciton transition in energy between A and B intralayer excitons, observable for T$=4 -300$ K, whereas no such transition is observed for the monolayer in the same structure in this energy range. The interlayer excitons consist of an electron localized in one layer and a hole state delocalized over the bilayer, which results in the unusual combination of high oscillator strength and a static dipole moment. We also find signatures of interlayer excitons involving the second highest valence band (B) and compare absorption calculations for different bilayer stackings. For homotrilayer MoS$_2$ we also observe interlayer excitons and an energy splitting between different intralayer A-excitons originating from the middle and outer layers, respectively.
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