Abstract: We present a classification of the possible quantum deformations of the supergroup GL(1|1) and its Lie superalgebra gl(1|1). In each case, the (super)commutation relations and the Hopf structures are explicitly computed. For each R matrix, one finds two inequivalent coproducts whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras. In the braided case, one recovers the classical algebra gl(1|1) for suitable limits of the deformation parameters but this is no longer true in the unbraided case.