A method for melting and/or refining metals and a cooling device for the graphite electrode used for the same are disclosed. Melting and/or refining of metal such as steel-making are performed in an electric arc furnace by energizing three graphite electrode sets each corresponding to each phase of a three-phase AC power source and consisting of a vertical succession of graphite electrodes as typically shown at 10 in FIGS. 2, 3 and 4, connected to one another via nipples.During the melting and/or refining of metal, a cooling liquid 11, substantially consisting of water for instance, is continuously blown against the outer periphery 10a of at least one of each set of graphite electrodes, specifically a graphite electrode 10 extending between an electrode holder and a furnace top cover. The liquid coolant 11 is jet not in the horizontal direction but in a downwardly or upwardly inclined direction at an angle of 10.degree. to 35.degree. C. with respect to the horizontal.