  We introduce a general notion of quantum universal enveloping algebroids (QUE
algebroids), or quantum groupoids, as a unification of quantum groups and
star-products. Some basic properties are studied including the twist
construction and the classical limits. In particular, we show that a quantum
groupoid naturally gives rise to a Lie bialgebroid as a classical limit.
  Conversely, we formulate a conjecture on the existence of a quantization for
any Lie bialgebroid, and prove this conjecture for the special case of regular
triangular Lie bialgebroids. As an application of this theory, we study the
dynamical quantum groupoid ${\cal D}\otimes_{\hbar} U_{\hbar}(\frakg)$, which
gives an interpretation of the quantum dynamical Yang-Baxter equation in terms
of Hopf algebroids.
