  Unbounded order convergence has lately been systematically studied as a
generalization of almost everywhere convergence to the abstract setting of
vector and Banach lattices. This paper presents a duality theory for unbounded
order convergence. We define the unbounded order dual (or uo-dual)
$X_{uo}^\sim$ of a Banach lattice $X$ and identify it as the order continuous
part of the order continuous dual $X_n^\sim$. The result allows us to
characterize the Banach lattices that have order continuous preduals and to
show that an order continuous predual is unique when it exists. Applications to
the Fenchel-Moreau duality theory of convex functionals are given. The
applications are of interest in the theory of risk measures in Mathematical
Finance.
