In an eyeglass lens processing apparatus for beveling a peripheral edge of an eyeglass lens, if the high curve lens processing mode is selected by a mode selector, a computing unit acquires a high curve bevel path for locating the bevel apex on a front surface curve of the eyeglass lens or for locating the bevel apex at a position shifted by a predetermined quantity from the front surface curve toward the rear side on the basis of the edge position information acquired by the edge position detector, thereby providing high curve beveling data for the rear surface beveling grindstone, or for the front surface and rear surface beveling grindstones; and a beveling controller bevels the peripheral edge of the eyeglass lens by the rear surface beveling grindstone, or by the front surface and rear surface beveling grindstones on the basis of the high curve beveling data.