High-pressure experiments on diffusion in MgO were performed to model the rheological and chemical transport properties of the lower mantle. Lattice and grain boundary diffusion coefficients for Mg, O and Al were determined at 2273 K and pressures up to 25 GPa. The results for pure MgO are in excellent agreement with first-principles calculations. In samples doped with Al2O3, cation vacancies were found to attach to Al impurities with a binding energy of ~0.7 eV. Our results predict that a transition from diffusion creep to dislocation creep will occur in the deep lower mantle if the shear stress exceeds ~1--10 MPa, for a grain size of ~0.1--1 mm. Diffusion through periclase is fast enough to allow substantial chemical exchange across the core-mantle boundary since core formation, with length scales of ~1--10 km for lattice diffusion and 100 km for grain boundary diffusion.