Abstract: This invention relates to a tumbling mill which has a generally polygonal liner, which defines a grinding chamber. The cross-section of the grinding chamber is defined by liner plates and is generally polygonal and preferably approximately square. At the corners, the liner plates are directly supported on the inside surface of a shell which surrounds the liner. Elsewhere, the liner plates are supported on the inside surface by spacers. The liner plates extend along elliptic arcs on the inside of the shell.