NOVELTY - The method for producing biodiesel in the form of a fatty acid alkyl ester by esterifying vegetable/animal oils-and-fats/wastes cooking oils, and with supercritical alcohols, comprises pumping the alcohol and the oils-and-fats under a pressure into a mixer (107) in which they are uniformly mixed with each other, elevating the temperature of the mixture through a heat exchanger, heating the temperature-elevated mixture to a predetermined temperature, and subjecting the mixture to the esterification conditions where the alcohol is maintained at a supercritical state.    USE - Useful for producing biodiesel in the form of fatty acid alkyl ester by esterifying vegetable/animal oils-and-fats/wastes cooking oils, and with supercritical alcohols.    ADVANTAGE - The method produces high-purity fatty acid alkyl ester at low cost and high productivity without using catalyst.    DETAILED DESCRIPTION - The method for producing biodiesel in the form of a fatty acid alkyl ester by esterifying vegetable/animal oils-and-fats/wastes cooking oils, and with supercritical alcohols, comprises pumping the alcohol and the oils-and-fats under a pressure into a mixer (107) in which they are uniformly mixed with each other, elevating the temperature of the mixture through a heat exchanger, heating the temperature-elevated mixture to a predetermined temperature, subjecting the mixture to the esterification conditions where the alcohol is maintained at a supercritical state, reducing the pressure of the heat-exchanged product to separate and recover the alcohols, separating and recovering the fatty acid alkyl ester from the reaction product from which the alcohols is removed, purifying the recovered biodiesel, and removing dissolved oxygen present in each of the oils-and-fats and the alcohol. The content of the recovered fatty acid alkyl ester does not reach a predetermined level. The supercritical esterification is additionally conducted more than one time. Before pumping the alcohol into the mixer, the alcohol above the critical point through the separate heat exchanger is maintained. The heat exchanger is a cross heat exchanger, or an exchanger of conducting heat exchange using an external heat medium. The heat exchanger uses the external heat medium, and comprises a heat exchanger network that has a function of controlling the esterification by increasing the amount of use of the heat medium. The oils-and-fats are soybean oil, rapeseed oil, sunflower seed oil, corn oil and/or palm oil. The alcohol has 1-8 carbon atoms and/or mixtures of two or more. The temperature (350-400degreesC) of the supercritical esterification is maintained above the critical temperature of the alcohols. The pressure (10-12 MPa) of the supercritical esterification is maintained above the critical pressure of the alcohols. The residence time of the reaction mixture in the supercritical esterification is 10-20 minutes. The volume of the alcohols used in the esterification is 0.5-2 times the volume of the oils-and-fats. The conditions of a first-order esterification are different from those of a second-order or higher-order esterification. The recovered alcohol is recycled as a raw material.    DESCRIPTION OF DRAWING(S) - The diagram shows a system for producing biodiesel:    Storage tank; (101)    Oxygen-removing unit; (102)    Pressurizing pumps; (103, 106)    Mixer; (107)    First heat exchanger. (201)