Climbing equipment for the construction of a building, or building under construction , said building under construction comprising a plurality of building floors , wherein said climbing equipment comprises at least one scaffold , adapted to surround at least partially the perimeter of at least one building floor ; and wherein said scaffold comprises at least one work platform , adapted to form a walkable surface and adapted to project like a cantilever from said scaffold towards said building under construction , wherein at least some of said at least one work platform being adapted to close the open space between said scaffold and said building under construction ; and wherein said climbing equipment further comprises a traction device comprising at least a first portion integral with said scaffold and at least a second portion connected to said first portion ; wherein said at least a second portion is slidingly connected to said first portion ; and wherein said at least a first portion comprises at least a first coupling element and said at least a second portion comprises at least a second coupling element ; and wherein said first coupling element is adapted to grip a first coupling portion of said building under construction , and wherein said second coupling element is adapted to grip a second coupling portion of said building under construction ; and wherein said first coupling portion is arranged at a different level with respect to said second coupling portion ; at least either said first coupling portion or said second coupling portion comprises a coupling pin , adapted to be accommodated in a hole made in an upright , of the building under construction ; and wherein said climbing equipment further comprises a moving system , adapted to perform a relative movement between said first portion and said second portion of said traction device ; and wherein when said at least a first coupling element is released from the respective coupling portion of said building under construction , said moving system is adapted to perform a relative movement between said first portion and said second portion of said traction device , so as to move said scaffold by a predetermined movement defining a direction of relative movement (X); and wherein the extension of said scaffold parallel to the direction of relative movement (X) is greater than said first portion of the traction device , which is integral with said scaffold , by a predetermined amount , at least equal to the height of one building floor ; characterized in that said first coupling element is rotationally connected to said first portion of said traction device and wherein said first coupling element is adapted to project cantilevered from said first portion of said traction device , like a shelf; and said second coupling element is rotationally connected to said second portion of said traction device ; and wherein said second coupling element is adapted to project cantilevered from said second portion of said traction device , like a shelf; and in that said first coupling element and said second coupling element each comprise at least one hook , so that said at least one hook is adapted to be releasably coupled to said coupling pin included in said building under construction . <EOS>