A modular disc prosthesis system is provided that allows a disc prosthesis to be constructed, either pre- or intra-operatively, to have a desired degree of freedom or constraint when positioned between endplates of adjacent vertebrae. In particular, the components can be assembled to form a disc prosthesis having a center of rotation that is constrained or fixed such that, when the disc prosthesis is implanted between endplates of adjacent vertebrae, the adjacent vertebrae have one degree of freedom and move about a fixed center of rotation. Alternatively, one or more of the same components and one or more additional components of the system can be used to construct a disc prosthesis having a center of rotation that is unconstrained or that floats such that, when the disc prosthesis is implanted between the endplates of adjacent vertebrae, the adjacent vertebrae have multiple degrees of freedom and can move about a floating or moving center of rotation, i.e., a center of rotation that is not fixed.