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 . <EOS>