An intersomatic cage, an intervertebral prosthesis, an anchoring device and an instrument for implantation of the cage or the prosthesis and the anchoring device are provided. An intersomatic cage or an intervertebral prosthesis fit closely to the anchoring device, which includes a body of elongated shape on a longitudinal axis, of curved shape describing, along the longitudinal axis, an arc whose dimensions and radius of curvature are designed in such a manner that the anchoring device may be implanted in the vertebral plate of a vertebra by presenting its longitudinal axis substantially along the plane of the intervertebral space, where the anchoring device is inserted, by means of the instrument, through a slot located in at least one peripheral wall of the cage or on at least one plate of the intervertebral disc prosthesis to penetrate into at least one vertebral plate.