Abstract: An object of the present invention is to provide a process for continuously casting a molten metal which process suppresses the instability of the initial solidification and stably improves the lubrication and the surface properties of the cast metal, and an apparatus therefore, in the process for continuously casting a molten metal an alternating current is applied to an electromagnetic coil which is provided so that it surrounds a continuous casting mold wall or is embedded in the side wall of the mold, whereby an electromagnetic force is exerted on the molten metal poured into the mold which either oscillates in a constant mode or does not oscillate and is starting to be solidified the process of the present invention also comprises periodically changing the amplitude or waveform of the alternating current to be applied, and the apparatus of the present invention is used for the process.