The effect of thermal annealing on the surface morphology of thin gold films is studied using a scanning tunneling microscope (STM) and computer simulations. The gold films were thermally evaporated onto glass substrates, and were then measured with the STM at room temperature before and after annealing. The annealing treatments were done at temperatures between 200 and 500 degrees C and for periods ranging from 1 to 200 h. We present data showing the evolution of the average surface-grain size and root-mean-square roughness amplitude of the gold films as a function of annealing temperature and duration. Our data suggest that surface diffusion is the main process active at low annealing temperatures of 300 degrees C and below. At higher annealing temperatures grain coarsening, which can be explained by recovery and recrystallization (secondary grain growth), is the dominant process contributing to large scale morphology changes. Computer simulations based on these processes account well for the experimental results, with activation energies of about 1.0 and 1.4 eV, for surface self-diffusion of gold and grain coarsening, respectively.