A process for producing oxygen by using of three-stage pressure swing adsorption plants, wherein the process is used to separate nitrogen and oxygen from a feed air stream, the product can be oxygen or nitrogen or both of them. The process utilizes three-stage pressure swing adsorption plants which are serially connected. In the first stage, carbon dioxide, water and part of nitrogen are removed and nitrogen is concentrated. In the second stage, nitrogen is further separated from the effluent intermediate gas from the adsorption step in the adsorption towers of the first stage and oxygen is concentrated to the desired concentration. In the third stage, nitrogen and argon are further separated from the effluent oxygen-enriched mixture gas from the adsorption step in the adsorption towers of the second stage and the concentration of oxygen is raised to 95V % or more.