Jean-Michel Vallin had put into light the existence, in the case of groupoids, of a unitary which plays the role of the multiplicative unitary of a locally compact group. A non-commutative analog had been then founded by J.-M. Vallin and the author in depth 2 inclusions of von Neumann algebras. These examples are presented, and a general theory of pseudo-multiplicative unitaries is developped, showing that this notion is a good candidate for a definition of "quantum groupoid".