At the low temperatures of interstellar dust grains, it is well established that surface chemistry proceeds via diffusive mechanisms of H atoms weakly bound (physisorbed) to the surface. Until recently, however, it was unknown whether atoms heavier than hydrogen could diffuse rapidly enough on interstellar grains to react with other accreted species. In addition, models still require simple reduction as well as oxidation reactions to occur on grains to explain the abundances of various molecules. In this paper we investigate O-atom diffusion and reactivity on a variety of astrophysically relevant surfaces (water ice of three different morphologies, silicate, and graphite) in the 6.5 – 25 K temperature range. Experimental values were utilized to derive a diffusion law which emphasizes that O-atoms diffuse by quantum mechanical tunnelling at temperatures as low as 6.5 K. The rate of diffusion on each surface, based on modeling results, were calculated and an empirical law is given as a function of the surface temperature. Relative diffusion rates are k_H2Oice > k_sil > k_grap >> k_expected. The implications of an efficient O-atom diffusion over astrophysically relevant time-scales are discussed. Our findings show that O atoms can scan any available reaction partners (e.g., either another H atom, if available, or a surface radical like O or OH) at a faster rate than that of accretion. Also, as dense clouds mature H2 becomes far more abundant than H and the O/H ratio grows, the reactivity of O atoms on grains is such that O becomes one of the dominant reactive partners together with H.