  We introduce C*-pseudo-multiplicative unitaries and concrete Hopf
C*-bimodules for the study of quantum groupoids in the setting of C*-algebras.
These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and
Hopf C*-algebras and are analogues of the pseudo-multiplicative unitaries and
Hopf--von Neumann-bimod-ules studied by Enock, Lesieur and Vallin. To each
C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a
duality pairing, a C*-tensor category of representations, and in the regular
case two reduced and two universal Hopf C*-bimodules. The theory is illustrated
by examples related to locally compact Hausdorff groupoids. In particular, we
obtain a continuous Fourier algebra for a locally compact Hausdorff groupoid.
