Simultaneous high-pressure Brillouin spectroscopy and powder X-ray diffraction of cerium dioxide powders are presented at room temperature to a pressure of 45 GPa. Micro- and nanocrystalline powders are studied and the density, acoustic velocities and elastic moduli determined. In contrast to recent reports of anomalous compressibility and strength in nanocrystalline cerium dioxide, the acoustic velocities are found to be insensitive to grain size and enhanced strength is not observed in nanocrystalline CeO2. Discrepancies in the bulk moduli derived from Brillouin and powder X-ray diffraction studies suggest that the properties of CeO2 are sensitive to the hydrostaticity of its environment. Our Brillouin data give the shear modulus, G0 = 63 (3) GPa, and adiabatic bulk modulus, KS0 = 142 (9) GPa, which is considerably lower than the isothermal bulk modulus, KT0∼ 230 GPa, determined by high-pressure X-ray diffraction experiments.