  The purpose of this paper is to give a general survey of Hom-bialgebras,
which are bialgebra-type structures where the identities are twisted by a
morphism, and to extend the concept of quasi-bialgebra to Hom-setting. We
provide some key constructions and generalize the concept of gauge
transformation and Drinfeld's twists to this type of generalized
quasi-bialgebras. Moreover, we provide an example of twisted quantum double.
