An apparatus for covering a box which has a bottom wall with more than four sides and a lateral wall having as many faces, using a flat covering blank which has a base shaped like the bottom wall of the box and segments connected to the base which can be folded to cover the lateral wall of the box, at least one oversized segment of the segments being made in such a way that it completely covers at least two of the faces of the lateral wall of the box and partly covers at least another face of the lateral wall, the apparatus comprising: - means for placing the box in a folding station, on the pre-glued covering blank positioned on a horizontal surface, with the bottom wall in contact with the base of the blank; - a mould which moves vertically and is operatively connected to an inner surface of the box to move it vertically from the folding station to a fold over station which is lower than the folding station; - at least one contact element positioned in the folding station for interacting with said at least an oversized segment of the covering blank during its motion as one with the box, causing the oversized segment to be folded from the horizontal position to a vertical position in which it is in contact with a corresponding face of the lateral wall of the box; - at least two folder elements movable relative to the box positioned in the fold over station; - movement means connected to the folder elements for laterally folding said at least one oversized segment and bringing it completely into contact with the lateral wall of the box, the apparatus being characterized in that the movement means of at least one of the folder elements are adapted to move it along a predetermined path comprising at least a first straight stretch of predetermined length for folding the oversized segment to completely cover another face of the lateral wall and a second stretch which is at a predetermined angle to the first stretch for folding the oversized segment to partly cover further another face of the lateral wall, said predetermined path comprising at least one change of direction correspondent to said predetermined angle such that the folder element follows the profile of the lateral wall of the box. - means for placing the box in a folding station, on the pre-glued covering blank positioned on a horizontal surface, with the bottom wall in contact with the base of the blank; - a mould which moves vertically and is operatively connected to an inner surface of the box to move it vertically from the folding station to a fold over station which is lower than the folding station; - at least one contact element positioned in the folding station for interacting with said at least an oversized segment of the covering blank during its motion as one with the box, causing the oversized segment to be folded from the horizontal position to a vertical position in which it is in contact with a corresponding face of the lateral wall of the box; - at least two folder elements movable relative to the box positioned in the fold over station; - movement means connected to the folder elements for laterally folding said at least one oversized segment and bringing it completely into contact with the lateral wall of the box,