Equipment for covering folding straight paperboard or microwave carton boxes by means of cold-sticker sheet portions, each box comprising a bottom and a body which rises from the bottom defining an outer lateral surface and a free edge opposite to said bottom, said cold-sticker sheet portions comprising lateral surface covering portions and bottom covering portions; the equipment comprising a carousel rotating about a carousel axis of rotation, supporting a plurality n of molds each for the support of a respective box, with said bottom and said free edge in correspondence respectively to an upper surface and to a base of the mold, said molds being equally distanced from said axis of rotation and equally spaced along a periphery of the carousel, each mold being pivotally supported by said carousel around a mold axis (Z') parallel to said carousel axis of rotation, the equipment further comprising driving means for driving the rotation of the carousel in angular steps each of 360°/n, in each step said molds occupying respective stations, said stations comprising at least: - a loading and unloading station, for unloading a covered box and loading a box to be covered, or box blanks corresponding respectively to said body and bottom; - a bottom covering station, comprising first peeling means for peeling individual bottom covering portions from a first sticker portion liner feed, and suction means adapted to catch said individual bottom covering portions from an outlet of said first peeling means and to release them onto sticking contact with said bottom; - a lateral covering station for covering the outer lateral surface of the body, comprising first mold rotation drive means adapted to drive the mold into rotation about its mold axis (Z'), and second peeling means for peeling individual lateral covering portions from a second sticker portion liner feed, adapted to feed the said lateral covering portions in tangential relationship with the mold, until sticking contact is made between a head flap of the lateral covering portion and said lateral surface of said body, whereby as a result effect of a coordinated rotation of the mold said lateral covering portion is wound in adhesion onto said lateral surface of said body; - and a covering tucking station for tucking the lateral covering portion around said free edge of said body, comprising second mold rotation drive means adapted to drive the mold into rotation about its mold axis (Z'), and a tucking device comprising a movable skid, movable towards and away from said mold and adapted to strike an excess flap (RLe) of said lateral covering portion protruding beyond said free edge of said body, whereby as a result of a coordinated rotation of the mold said excess portion (RLe) is tucked into sticking contact around said free edge. - a loading and unloading station, for unloading a covered box and loading a box to be covered, or box blanks corresponding respectively to said body and bottom; - a bottom covering station, comprising first peeling means for peeling individual bottom covering portions from a first sticker portion liner feed, and suction means adapted to catch said individual bottom covering portions from an outlet of said first peeling means and to release them onto sticking contact with said bottom; - a lateral covering station for covering the outer lateral surface of the body, comprising first mold rotation drive means adapted to drive the mold into rotation about its mold axis (Z'), and second peeling means for peeling individual lateral covering portions from a second sticker portion liner feed, adapted to feed the said lateral covering portions in tangential relationship with the mold, until sticking contact is made between a head flap of the lateral covering portion and said lateral surface of said body, whereby as a result effect of a coordinated rotation of the mold said lateral covering portion is wound in adhesion onto said lateral surface of said body; - and a covering tucking station for tucking the lateral covering portion around said free edge of said body, comprising second mold rotation drive means adapted to drive the mold into rotation about its mold axis (Z'), and a tucking device comprising a movable skid, movable towards and away from said mold and adapted to strike an excess flap (RLe) of said lateral covering portion protruding beyond said free edge of said body, whereby as a result of a coordinated rotation of the mold said excess portion (RLe) is tucked into sticking contact around said free edge.