The monomolecular surface layer of acceptor doped CeO₂ may become neutral and metallic or charged and semiconducting. This is revealed in the theoretical analysis of the oxygen pressure dependence of the surface defects concentration in acceptor doped ceria with two different dopant types and operated under different oxygen pressures. Recently published experimental data for highly reduced Sm_0.2Ce_0.8O_1.9–x (SDC) containing a fixed valence dopant Sm³⁺ are very different from those published for Pr_0.1Ce_0.9O_2–x (PCO) with the variable valence dopant Pr⁴⁺/Pr³⁺ being reduced under milder conditions. The theoretical analysis of these experimental results fits very well the experimental results of SDC and PCO. It leads to the following predictions: the highly reduced surface of SDC is metallic and neutral, the metallic surface electron density of state is g_s = 0.9 × 10³⁸ J^–1·m^–2 (1.4 × 10¹⁵ eV^–1·cm^–2), the electron effective mass is m_eff,s = 3.3m_e, and the phase diagram of the reduced surface has the α (fcc) structure as in the bulk. In PCO a double layer is predicted to be formed between the surface and the bulk with the surface being negatively charged and semiconducting. The surface of PCO maintains high Pr³⁺ defect concentration as well as relative high oxygen vacancy concentration at oxygen pressures higher than in the bulk. The reasons for the difference between a metallic and semiconducting surface layer of acceptor doped CeO₂ are reviewed, as well as the key theoretical considerations applied in coping with this problem. For that we make use of the experimental data and theoretical analysis available for acceptor doped ceria.