Device for mounting a handle on a leaf of a door or window, wherein said handle comprises a main body defining a handgrip and a pin having polygonal cross section engageable with a moving device for moving opening and closing members of said door or window, wherein said moving device is housed in a recess of said leaf and is controlled by said pin, comprising: - a base body that is provided with connection members for the rigid, complete and removable connection of said base body at a leaf of a door or window, - a bush that is supported by said base body in a rotary manner around a rotation axis coinciding with the longitudinal axis of said bush, wherein said bush is crossed by a through hole that is coaxial to said rotation axis and that, along at least one axial segment thereof, has polygonal cross section substantially matching that of said pin, - first axial locking members for locking the relative sliding of said base body and of said bush along said rotation axis, and - snap-stop means for removably stopping said bush in at least two different angular positions and that are arranged between said base body and said bush, wherein said device can be fixed to a wall of said leaf at a slot, said slot being formed in said wall at said recess, with said rotation axis substantially orthogonal to said wall, and wherein said bush can be crossed by said pin, the mutually axially opposite end segments of which - a first axial end segment and a second axial end segment, respectively - project from the opposite faces of said device and respectively engage in said moving device and in said main body of said handle, an intermediate axial segment of said pin arranged between said end segments thereof coupling with a prismatic coupling with said bush, characterized in that - said connection members comprise at least one pair of clamping plates that are supported by said base body by means of respective screws, wherein said clamping plates can be inserted in said slot below said wall to clamp said wall between them and said base body, wherein each of said clamping plates is guided in a corresponding seat formed in said base body and configured so as to prevent the rotation of the respective clamping plate together with the corresponding screw. - a base body that is provided with connection members for the rigid, complete and removable connection of said base body at a leaf of a door or window, - a bush that is supported by said base body in a rotary manner around a rotation axis coinciding with the longitudinal axis of said bush, wherein said bush is crossed by a through hole that is coaxial to said rotation axis and that, along at least one axial segment thereof, has polygonal cross section substantially matching that of said pin, - first axial locking members for locking the relative sliding of said base body and of said bush along said rotation axis, and - snap-stop means for removably stopping said bush in at least two different angular positions and that are arranged between said base body and said bush, wherein said device can be fixed to a wall of said leaf at a slot, said slot being formed in said wall at said recess, with said rotation axis substantially orthogonal to said wall, and wherein said bush can be crossed by said pin, the mutually axially opposite end segments of which - a first axial end segment and a second axial end segment, respectively - project from the opposite faces of said device and respectively engage in said moving device and in said main body of said handle, an intermediate axial segment of said pin arranged between said end segments thereof coupling with a prismatic coupling with said bush, characterized in that - said connection members comprise at least one pair of clamping plates that are supported by said base body by means of respective screws, wherein said clamping plates can be inserted in said slot below said wall to clamp said wall between them and said base body, wherein each of said clamping plates is guided in a corresponding seat formed in said base body and configured so as to prevent the rotation of the respective clamping plate together with the corresponding screw.