An eyeglass lens grinding apparatus, which performs bevelling on an eyeglass lens while sufficiently reducing the variation in the size of the bevel being formed so that the finished lens can be fitted snugly in the wearer's eyeglass frame. The eyeglass lens grinding apparatus includes a bevel position determining system for determining the position of the apex of a bevel to be formed on the lens being processed, a bevelling abrasive wheel that has a first inclined bevelling surface and a second inclined bevelling surface and which processes the front and rear surfaces of the bevel independently of each other, a lens rotating shaft that holds and rotates the lens, a bevel calculating system that determines the processing points at which said first and second inclined bevelling surfaces process the lens and which determines two kinds of bevelling data, one for processing the front surface of the bevel and the other for processing its rear surface in such a way that said apex of the bevel being formed contacts said first and second inclined bevelling surfaces in correspondence with the thus determined processing points, and a bevelling controller that controls the bevelling operation on the basis of the two kinds of bevelling data as determined by said bevel calculating system.