In recent years due to improvements in calculation methods and increased computer power, it has become possible to perform first-principles investigations for "simple" chemical reactions at surfaces. We have carried out such studies for the catalytic oxidation of CO at transition metal surfaces, in particular, at the ruthenium surface for which unusual behavior compared to other transition metal catalysts has been reported. High gas pressure catalytic reactor experiments have revealed that the reaction rate over Ru for oxidizing conditions is the highest of the transition metals considered -in contrast, under ultra high vacuum conditions, the rate is by far the lowest. We find that important for understanding the pressure dependence of the reaction is the fact that Ru (0001) can support high concentrations of oxygen at the surface. Under these conditions, the O-metal bond is atypically weak compared to that at lower coverages. We have investigated a number of possible reaction pathways for CO oxidation for the conditions of high oxygen coverages, including scattering reactions of gas-phase CO at the oxygen covered surface (Eley-Rideal mechanism) as well as the Langmuir-Hinshelwood mechanism involving reaction between adsorbed CO molecules and O atoms.PACSnumbers: 82.65.J, 71.10, 82.65.M