An indexable cutting insert for a cutting tool, comprising: - a first side surface and an opposing second side surface, and a peripheral surface extending between the first side surface and the second side surface; - a cutting portion comprising a first cutting face and an opposing second cutting face for facing in the cutting direction during use, a flank face extending between the first cutting face and the second cutting face, a first cutting edge between the first cutting face and the flank face, and a second cutting edge between the second cutting face and the flank face; - wherein the cutting portion is bisected by a first plane (AA') transverse to the first side surface; and - wherein the cutting portion has a cross-section transverse to the first plane (AA') that is concave on both the first side surface and on the second side surface, - wherein, on the cutting portion, the first side surface is concave and defines a portion of a surface of a first cylinder having its axis in the first plane, - wherein, on the cutting portion, the second side surface is concave and defines a portion of a surface of a second cylinder having its axis (CC') in the first plane (AA'), and wherein the first cylinder and the second cylinder are elliptic cylinders, each of which having a transverse cross-section that forms an ellipse, wherein the cross-section has a semi-major axis that is transverse to the first plane, and wherein the axis of the first cylinder and the axis of the second cylinder diverge towards the flank face, such that a width of the cutting portion between the first side surface and the second side surface increases towards the flank face. - a first side surface and an opposing second side surface, and a peripheral surface extending between the first side surface and the second side surface; - a cutting portion comprising a first cutting face and an opposing second cutting face for facing in the cutting direction during use, a flank face extending between the first cutting face and the second cutting face, a first cutting edge between the first cutting face and the flank face, and a second cutting edge between the second cutting face and the flank face; - wherein the cutting portion is bisected by a first plane (AA') transverse to the first side surface; and - wherein the cutting portion has a cross-section transverse to the first plane (AA') that is concave on both the first side surface and on the second side surface, - wherein, on the cutting portion, the first side surface is concave and defines a portion of a surface of a first cylinder having its axis in the first plane, - wherein, on the cutting portion, the second side surface is concave and defines a portion of a surface of a second cylinder having its axis (CC') in the first plane (AA'), and wherein the first cylinder and the second cylinder are elliptic cylinders, each of which having a transverse cross-section that forms an ellipse, wherein the cross-section has a semi-major axis that is transverse to the first plane, and wherein the axis of the first cylinder and the axis of the second cylinder diverge towards the flank face, such that a width of the cutting portion between the first side surface and the second side surface increases towards the flank face.