In this paper we investigate composition models of incarnation, according to which Christ is a compound of qualitatively and numerically different constituents. We focus on three-part models, according to which Christ is composed of a divine mind, a human mind, and a human body. We consider four possible relational structures that the three components could form. We argue that a ‘hierarchy of natures’ model, in which the human mind and body are united to each other in the normal way, and in which they are jointly related to the divine mind by the relation of co-action, is the most metaphysically plausible model. Finally, we consider the problem of how Christ can be a single person even when his components may be considered persons. We argue that an Aristotelian metaphysics, according to which identity is a matter of function, offers a plausible solution: Christ's components may acquire a radically new identity through being parts of the whole, which enables them to be reidentified as parts, not persons.