We present a detailed ab initio investigation of the stability, the structural, electronic, and magnetic properties of the (0001) surfaces of hematite (Fe₂O₃) and chromia or eskolaite (Cr₂O₃). Strong electron correlation effects not included in a density-functional description are described by a Hubbard-type on-site Coulomb repulsion (the DFT+U approach). For bulk chromia we find, complementing our recent work on hematite [Rollmann et al. Phys. Rev. B 69 165107 (2004)] that the inclusion of correlation effects leads to an improved description of the structural, electronic, and magnetic properties. In particular, the increased exchange splitting of the d band changes the character of the insulating gap from a pure d-d Mott-Hubbard type to intermediate between d-d and charge-transfer insulator. For both oxides, the strong correlation effects have a dramatic influence on the surface stability: oxygen-terminated surfaces are strongly disfavored because of the increased energetic cost of stabilizing a higher oxidation state of the transition metal close to the surface. The stability of metal-terminated surfaces even under oxidizing conditions agrees with the most recent STM and LEED data. For Cr2O3(0001) where detailed experimental information on the surface structure is available, quantitative agreement of the calculated surface relaxations is achieved. Detailed results on the surface electronic structure (valence-band spectra and core-level shifts) and the surface magnetic properties are presented.