  We examine various versions of oriented cohomology and Borel-Moore homology
theories in algebraic geometry and put these two together in the setting of an
"oriented duality theory", a generalization of Bloch-Ogus twisted duality
theory. This combines and exends work of Panin and Mocanasu. We apply this to
give a Borel-Moore homology version $MGL'_{*,*}$ of Voevodsky's
$MGL^{*,*}$-theory, and a natural map $\vartheta:\Omega_*\to MGL'_{2*,*}$,
where $\Omega_*$ is the algebraic cobordism theory defined by Levine-Morel. We
conjecture that $\vartheta$ is an isomorphism and describe a program for
proving this conjecture.
