We report on the results of an atomistic, discrete-lattice calculation of positron surface states on the three principal surfaces of Al and Cu. We are able to (i) accurately reproduce the observed values and anisotropy of the binding energies, and (ii) predict the surface state life times. Furthermore, we calculate (iii) the positron lateral diffusion constant, and find it considerably enhanced over the bulk value. We also investigate (iv) the positron trapping at surface vacancies, and (v) the effect of ordered chemisorbed monolayers of oxygen. We find that the oxidation lowers the binding energy and makes the surface state unstable with respect to positronium emission on Al (100) and Al (111). Implications to surface studies are discussed.