The diffusion of vacancy islands on Cu(111) and Ag(111) and of adatom islands on Ag(111) has been studied using fast scanning STM. Diffusion of atoms along island edges (periphery diffusion) is much more effective in contributing to the diffusion of the islands than diffusion of atoms via terrace sites, under the conditions considered in this work. Assuming power-law scaling of the diffusion coefficient with the size of the islands, effective exponents, β, between 1.33 and 1.63 were determined. The effective diffusion barrier Ed was found to be 0.49 eV on Cu and between 0.51 and 0.53 eV on Ag, respectively. Activation energies for the diffusion of atoms along the island edges and for the breaking up of closed island edges (core breakup) were calculated using potentials derived from effective medium theory. Qualitative and quantitative comparisons between the measured and calculated activation energies for diffusion show that one cannot explain the island diffusion as being governed by one single rate-limiting process. The results are consistent with a mechanism for which both periphery diffusion and core breakup are rate-limiting for island diffusion.