  An equivalence between Lu's bialgebroids, Xu's bialgebroids with an anchor
and Takeuchi's $\times_{A}$-bialgebras is explicitly proven. A new class of
examples of bialgebroids is constructed. A (formal) dual of a bialgebroid,
termed bicoalgebroid, is defined. A weak Hopf algebra is shown to be an example
of such a bicoalgebroid.
