Gold-based catalysts have been of intense interests in recent years, being regarded as a new generation of catalysts due to their unusually high catalytic performance. For example, CO oxidation on Au/TiO(2) has been found to occur at a temperature as low as 200 K. Despite extensive studies in the field, the microscopic mechanism of CO oxidation on Au-based catalysts remains controversial. Aiming to provide insight into the catalytic roles of Au, we have performed extensive density functional theory calculations for the elementary steps in CO oxidation on Au surfaces. O atom adsorption, CO adsorption, O(2) dissociation, and CO oxidation on a series of Au surfaces, including flat surfaces, defects and small clusters, have been investigated in detail. Many transition states involved are located, and the lowest energy pathways are determined. We find the following: (i) the most stable site for O atom on Au is the bridge site of step edge, not a kink site; (ii) O(2) dissociation on Au (O(2)-->2O(ad)) is hindered by high barriers with the lowest barrier being 0.93 eV on a step edge; (iii) CO can react with atomic O with a substantially lower barrier, 0.25 eV, on Au steps where CO can adsorb; (iv) CO can react with molecular O(2) on Au steps with a low barrier of 0.46 eV, which features an unsymmetrical four-center intermediate state (O-O-CO); and (v) O(2) can adsorb on the interface of Au/TiO(2) with a reasonable chemisorption energy. On the basis of our calculations, we suggest that (i) O(2) dissociation on Au surfaces including particles cannot occur at low temperatures; (ii) CO oxidation on Au/inactive-materials occurs on Au steps via a two-step mechanism: CO+O(2)-->CO(2)+O, and CO+O-->CO(2); and (iii) CO oxidation on Au/active-materials also follows the two-step mechanism with reactions occurring at the interface.