Identity is a relation of equivalence, reflexive, symmetric and transitive, subject to Leibniz’s law and to the principle of indiscernibility of identicals. Could there be an identity relation which does not have these formal properties? It seems much harder to maintain that these properties apply when one considers the actual issues of the individuation of objects. Do the formal properties of identity apply in the same manner to spatial temporal continuity, to physical, to biological and to human entities ? This question has been debated from the Sophists onward. It has been proposed that identity be relativised to sortal concepts, that identity be replaced by a weaker relation of similarity, and some writers have accepted the claim that identity might be vague. The principle of indiscernibles, for example, has been said to fail for quantum mechanics. But strong arguments show, on the contrary, that the relation of identity cannot but be conform to classical principles. Drawing on Ruth Marcus pioneering work on essence, modality and identity, I try to examine in this paper, the following hypothesis : identity is a functional concept, or a second-order property of first-order properties of objects of any kind. In analogy to similar functionalist theories in the philosophy of mind and about truth, a functionalist conception of identity is a formal concept, realised in different ways in the kinds of entities to which it applies. I try to show that, whatever the merits of this functionalist conception, it nevertheless has strong limitations.