  We give a detailed comparison between the notion of a weak Hopf algebra (also
called a quantum groupoid by Nikshych and Vainerman), and that of a
$\times_R$-bialgebra due to Takeuchi (and also called a bialgebroid or quantum
(semi)groupoid by Lu and Xu). A weak bialgebra is the same thing as a
$\times_R$-bialgebra in which $R$ is Frobenius-separable. We extend the
comparison to cover module and comodule theory, duality, and the question when
a bialgebroid should be called a Hopf algebroid.
