Using first-principles calculations based on hybrid-density-functional theory, we examine the energetics and electronic structure of fluorine in α-Al2O3. The F atom can be incorporated as an interstitial (Fi) or substitutional impurity on the oxygen site (FO); the latter tends to be lower in energy, particularly under Al-rich conditions. Fluorine on the oxygen site acts as a donor, but for Fermi-level positions high in the bandgap, a negatively charged DX configuration is lower in energy. Fluorine substituting on the Al site is not energetically stable. We also examine complexes between F and hydrogen or carbon, which can easily be unintentionally incorporated during growth or processing. Our calculated defect levels, combined with band alignments, allow us to assess the impact on Al2O3/semiconductor heterostructures. We find that F can passivate oxygen-vacancy related traps in the Al2O3 dielectric. Complex formation with H or C is either ineffective or could even be detrimental.