Climbing equipment for the construction of a building, including a scaffold that at least partially surrounds the perimeter of a building floor and is provided with a walkable surface. The climbing equipment includes a traction device comprising a first portion integral with the scaffold, and a second portion connected to the first portion. The first portion includes a first coupling element adapted to grip a first coupling portion of the building, and the second portion includes a second coupling element, arranged at a different level than the first coupling element, and adapted to grip a second coupling portion of the building. The climbing equipment further comprises a moving system. When the first coupling element is released from the respective coupling portion, the moving system performs a relative movement between the first portion and said second portion of the traction device, so as to move the scaffold by a predetermined amount.