We report diffusion coefficients of micron-scale liquid domains in giant unilamellar vesicles of phospholipids and cholesterol. The trajectory of each domain is tracked, and the mean square displacement grows linearly in time, as expected for Brownian motion. We study domain diffusion as a function of composition and temperature and measure how diffusion depends on domain size. We find mechanisms of domain diffusion which are consistent with membrane-dominated drag in viscous L-o phases and bulk-dominated drag for less viscous L-alpha phases. Where applicable, we obtain the membrane viscosity and report activation energies of diffusion.