  Franck Lesieur had introduced in his thesis (now published in an expended and
revised version in the {\it M\'emoires de la SMF} (2007)) a notion of measured
quantum groupoid, in the setting of von Neumann algebras and a simplification
of Lesieur's axioms is presented in an appendix of this article. We here
develop the notions of actions, crossed-product, and obtain a biduality
theorem, following what had been done by Stefaan Vaes for locally compact
quantum groups. Moreover, we prove that the inclusion of the initial algebra
into its crossed-product is depth 2, which gives a converse of a result proved
by Jean-Michel Vallin and the author. More precisely, to any action of a
measured quantum groupoid, we associate another measured quantum groupoid. In
particular, starting from an action of a locally compact quantum group, we
obtain a measured quantum groupoid canonically associated to this action; when
the action is outer, this measured quantum groupoid is the initial locally
compact quantum group
