The graphene moiré superstructure formed on Ru(0001) (g/Ru(0001)) has shown the potential as a template to self-assemble super-lattices of metal nanoparticles as model catalysts. To explore the possibility of rational catalyst design on g/Ru(0001), detailed density functional theory (DFT) calculations have been performed to investigate the adsorption and diffusion of Rh and Au adatoms on g/Ru(0001). The consequences of different hopping rates for cluster nucleation have been explored by performing Monte Carlo-based statistical analysis, which suggests that diffusing species other than adatoms need to be taken into account to develop an accurate description of cluster nucleation and growth on this surface. DFT calculations have also been carried out to investigate the adsorption and diffusion of 18 4d (Y-Ag) and 5d (La-Au) transition metal adatoms on g/Ru(0001). Given the necessity to study larger diffusing species than adatoms, DFT calculations have been performed to study the adsorption and diffusion of Rh and Au dimers and trimers on g/Ru(0001). It was shown that the mobility of Rh clusters decreases with the increase of cluster size, while for Au, dimers diffuse faster than monomers and trimers on the moiré surface. We then used a genetic algorithm combined with DFT calculations to predict the lowest energy structure of a Au8 cluster on g/Ru(0001). Our prediction leads us to propose that Au clusters aggregates through Oswald ripening with Au dimer being the major diffusing species. Finally, we examined the morphology of a Cu19 cluster on g/Cu(111) using MD simulations with COMB3 potential. We also studied the mobility of Cu clusters on g/Cu(111) at elevated temperatures. The analysis suggests that g/Cu(111) may not be a suitable substrate for the formation and growth of isolated Cu clusters. All these calculation results have provided us a better understanding and useful insights into the nucleation and growth mechanism of metal clusters on graphene moiré.