A variable valve-operating device comprising a cam carrier in the form of a cylindrical member fitted around a cam shaft of a valve operating mechanism of an internal combustion engine so as to be inhibited from rotation relative to the cam shaft and slidable in an axial direction of the cam shaft, the cam carrier having a low speed side cam lobe and a high speed side cam lobe formed on an outer circumferential surface of the cam carrier to selectively act on an engine valve, the cam carrier also having lead grooves (4 and 5) formed in the outer circumferential surface of the cam carrier and including a speed increasing side lead groove and a speed decreasing side lead groove, the cam lobes (3L and 3H) having different cam profiles and being adjacent to each other in an axial direction of the cam carrier, and switching pins (8p and 9p) for engagement with the lead grooves (4 and 5), the switching pins (8p and 9p) including a speed increasing side switching pin and a speed decreasing side switching pin; the speed increasing side switching pin and the speed decreasing side switching pin being adapted to engageably and disengageably advance into and retreat from the speed increasing side lead groove and the speed decreasing side lead groove, respectively, and one of the cam lobes (3L and 3H) to selectively act on the engine valve being determined by axially shifting the cam carrier with one of the switching pins (8p and 9p) selectively advanced to engage with one of the lead grooves (4 and 5) of the cam carrier being rotated; wherein the speed increasing side lead groove with which the speed increasing side switching pin is engageable is operative for change from the low speed side cam lobe to the high speed side cam lobe and the speed decreasing side lead groove with which the speed decreasing side switching pin engages is operative for change from the high speed side cam lobe to the low speed side cam lobe; wherein the speed increasing side lead groove includes: a speed increasing side entry lead groove portion for accepting entry of the speed increasing side switching pin; a speed increasing switching lead groove portion extending from the speed increasing side entry lead groove portion to guide the cam carrier in the axial direction so as to move to a high speed side axial position; and a high speed steady position lead groove portion for enabling the cam carrier guided by the speed increasing switching lead groove portion to rotate steadily at the high speed side axial position; wherein the speed decreasing side lead groove includes: a speed decreasing side entry lead groove portion for accepting entry of the speed decreasing side switching pin; a speed decreasing switching lead groove portion extending from the speed decreasing side entry lead groove portion to guide the cam carrier in the axial direction so as to move to a low speed side axial position; and a low speed steady position lead groove portion for enabling the cam carrier guided by the speed decreasing switching lead groove portion to rotate steadily at the low speed side axial position; and wherein the speed increasing side entry lead groove portion is formed at a position axially overlapping the low speed steady position lead groove portion, and the speed decreasing side entry lead groove portion is formed at a position axially overlapping the high speed steady position lead groove portion. a speed increasing side entry lead groove portion for accepting entry of the speed increasing side switching pin; a speed increasing switching lead groove portion extending from the speed increasing side entry lead groove portion to guide the cam carrier in the axial direction so as to move to a high speed side axial position; and a high speed steady position lead groove portion for enabling the cam carrier guided by the speed increasing switching lead groove portion to rotate steadily at the high speed side axial position; wherein the speed decreasing side lead groove includes: a speed decreasing side entry lead groove portion for accepting entry of the speed decreasing side switching pin; a speed decreasing switching lead groove portion extending from the speed decreasing side entry lead groove portion to guide the cam carrier in the axial direction so as to move to a low speed side axial position; and a low speed steady position lead groove portion for enabling the cam carrier guided by the speed decreasing switching lead groove portion to rotate steadily at the low speed side axial position; and wherein the speed increasing side entry lead groove portion is formed at a position axially overlapping the low speed steady position lead groove portion, and the speed decreasing side entry lead groove portion is formed at a position axially overlapping the high speed steady position lead groove portion. a speed decreasing side entry lead groove portion for accepting entry of the speed decreasing side switching pin; a speed decreasing switching lead groove portion extending from the speed decreasing side entry lead groove portion to guide the cam carrier in the axial direction so as to move to a low speed side axial position; and a low speed steady position lead groove portion for enabling the cam carrier guided by the speed decreasing switching lead groove portion to rotate steadily at the low speed side axial position; and wherein the speed increasing side entry lead groove portion is formed at a position axially overlapping the low speed steady position lead groove portion, and the speed decreasing side entry lead groove portion is formed at a position axially overlapping the high speed steady position lead groove portion.