A turning insert includes a top surface, an opposite bottom surface, side surfaces connecting the top surface and the bottom surface and two opposite nose portions. Each nose portion includes a convex nose cutting edge, a first cutting edge and a second cutting edge. The convex nose cutting edge connects the first and second cutting edges. A bisector extends equidistantly from the first and second cutting edges. In a top view the first and second cutting edges on the same nose portion form a nose angle () of 70-85 relative to each other. Each nose portion includes a third convex cutting edge adjacent to the first cutting edge and a fourth cutting edge adjacent to the third cutting edge. In a top view the fourth cutting edge forms an angle () of 0-34 relative to the bisector.

A cutting insert of one aspect includes a top surface including a corner portion and a pair of side portions each adjacent to the corner portion, a bottom surface, a side surface located between the top surface and the bottom surface, and a corner cutting edge disposed at the corner portion. In such a cutting insert, the top surface includes a first rake surface located along the corner cutting edge, a raised portion located farther on the inner side than the first rake surface, and a second rake surface located between the corner cutting edge and the raised portion. Moreover, a rake angle at a first region of the second rake surface corresponding to a center of the corner cutting edge is greater than a rake angle at a second region of the second rake surface located beside the first region.

A straight line along the second line segment and a straight line along the fourth line segment form an acute angle. The straight line along the second line segment and the straight line along the third line segment form an obtuse angle. The straight line along the first line segment and the straight line along the fourth line segment form an obtuse angle. Each of the third curved portion and the fourth curved portion is larger in radius of curvature than the first curved portion and larger in radius of curvature than the second curved portion. In a direction perpendicular to the bottom surface, a distance between the top surface and the bottom surface is equal to a distance between the cutting edge and the bottom surface, or is shorter than the distance between the cutting edge and the bottom surface.

A turning insert includes a top surface, an opposite bottom surface, side surfaces connecting the top surface and the bottom surface and two opposite nose portions. Each nose portion includes a convex nose cutting edge, a first cutting edge and a second cutting edge. The convex nose cutting edge connects the first and second cutting edges. A bisector extends equidistantly from the first and second cutting edges. In a top view the first and second cutting edges on the same nose portion form a nose angle () of 70-85 relative to each other. Each nose portion includes a third convex cutting edge adjacent to the first cutting edge and a fourth cutting edge adjacent to the third cutting edge. In a top view the fourth cutting edge forms an angle () of 0-34 relative to the bisector.

A cutting insert has a top surface, a bottom surface, and an outer peripheral surface. A ridgeline between the top surface and the outer peripheral surface includes a first cutting edge. A protrusion is provided on the top surface. In a first cross section, a distance between the first cutting edge and the protrusion in a direction perpendicular to the bottom surface is defined as a first distance. In a second cross section, a distance between the first cutting edge and the protrusion in a direction perpendicular to the bottom surface is defined as a third distance. The third distance is longer than the first distance. A ratio of a height of the protrusion to a width of the protrusion in the second cross section is smaller than a ratio of a height of the protrusion to a width of the protrusion in the first cross section.