  We formulate an extension of the Calabi conjecture to the setting of
generalized K\"ahler geometry. We show a transgression formula for the Bismut
Ricci curvature in this setting, which requires a new local
Goto/Kodaira-Spencer deformation result, and use it to show that solutions of
the generalized Calabi-Yau equation on compact manifolds are classically
K\"ahler, Calabi-Yau, and furthermore unique in their generalized K\"ahler
class. We show that the generalized K\"ahler-Ricci flow is naturally adapted to
this conjecture, and exhibit a number of a priori estimates and monotonicity
formulas which suggest global existence and convergence. For initial data in
the generalized K\"ahler class of a K\"ahler Calabi-Yau structure we prove the
flow exists globally and converges to this unique fixed point. This has
applications to understanding the space of generalized K\"ahler structures, and
as a special case yields the topological structure of natural classes of
Hamiltonian symplectomorphisms on hyperK\"ahler manifolds. In the case of
commuting-type generalized K\"ahler structures we establish global existence
and convergence with arbitrary initial data to a K\"ahler, Calabi-Yau metric,
which yields a new $d d^c$-lemma for these structures.
