An adjustable roller unit for moving a slat of an aircraft wing, the slat being fixed to a plurality of rails, each rail sliding on facing pairs of roller units carried by facing pairs of ribs (11, 12) between a closed slat position and a deployed slat position, wherein each roller unit includes: a roller supporting pin having a stem portion capable of being fixed in an adjustable manner to one rib (11, 12) of the wing and defining a first axis (y); a roller mounted idle on the roller supporting pin and rotatable around a second axis (x) that is eccentric with respect to the first axis (y); a first threaded nut for locking the stem portion of the roller supporting pin in a selected angular position to place the second axis (x) of the roller at a selected relative height with respect to the first axis (y); adjusting means for adjusting an axial position of the roller supporting pin with respect to said rib (11, 12) and therefore also adjust consequently an axial gap between the roller and the associated rail of a slat; wherein the means for adjusting the axial position of the roller supporting pin comprise: a threaded length of the stem portion of the roller supporting pin and a threaded annular element cooperating with the threaded length to define therewith a desired axial position of the roller; an axially splined end length of the stem portion of the roller supporting pin and an anti-rotation annular element axially slidingly coupled with the splined end length of the stem portion, the anti-rotation annular element engaging the threaded annular element to prevent rotation of the threaded annular element on the threaded length of the stem portion; the roller unit being characterized in that in the desired axial position of the roller, the threaded annular element is axially abutting against the first threaded nut; and in that the means for adjusting the axial position of the roller supporting pin further comprise a second threaded locking nut for axially tightening the anti-rotation element against the threaded annular element and providing an axial abutment for the threaded annular element and the anti-rotation element on an axial side opposite to the first threaded nut. a roller supporting pin having a stem portion capable of being fixed in an adjustable manner to one rib (11, 12) of the wing and defining a first axis (y); a roller mounted idle on the roller supporting pin and rotatable around a second axis (x) that is eccentric with respect to the first axis (y); a first threaded nut for locking the stem portion of the roller supporting pin in a selected angular position to place the second axis (x) of the roller at a selected relative height with respect to the first axis (y); adjusting means for adjusting an axial position of the roller supporting pin with respect to said rib (11, 12) and therefore also adjust consequently an axial gap between the roller and the associated rail of a slat; wherein the means for adjusting the axial position of the roller supporting pin comprise: a threaded length of the stem portion of the roller supporting pin and a threaded annular element cooperating with the threaded length to define therewith a desired axial position of the roller; an axially splined end length of the stem portion of the roller supporting pin and an anti-rotation annular element axially slidingly coupled with the splined end length of the stem portion, the anti-rotation annular element engaging the threaded annular element to prevent rotation of the threaded annular element on the threaded length of the stem portion; the roller unit being characterized in that in the desired axial position of the roller, the threaded annular element is axially abutting against the first threaded nut; and in that the means for adjusting the axial position of the roller supporting pin further comprise a second threaded locking nut for axially tightening the anti-rotation element against the threaded annular element and providing an axial abutment for the threaded annular element and the anti-rotation element on an axial side opposite to the first threaded nut. a threaded length of the stem portion of the roller supporting pin and a threaded annular element cooperating with the threaded length to define therewith a desired axial position of the roller; an axially splined end length of the stem portion of the roller supporting pin and an anti-rotation annular element axially slidingly coupled with the splined end length of the stem portion, the anti-rotation annular element engaging the threaded annular element to prevent rotation of the threaded annular element on the threaded length of the stem portion; the roller unit being characterized in that in the desired axial position of the roller, the threaded annular element is axially abutting against the first threaded nut; and in that the means for adjusting the axial position of the roller supporting pin further comprise a second threaded locking nut for axially tightening the anti-rotation element against the threaded annular element and providing an axial abutment for the threaded annular element and the anti-rotation element on an axial side opposite to the first threaded nut.