  In the present paper, we consider several valid notions of orientability of
Alexandov spaces and prove that all such conditions are equivalent. Further, we
give topological and geometric applications of the orientability. In
particular, a Poincar\'e-type duality theorem is proved. As a corollary to the
duality theorem, we also prove that if a closed Alexandrov space admits a
positive curvature bound in a synthetic sense, then its codimension one
homology vanishes. Further, we obtain a filling radius inequality for closed
orientable Alexandrov spaces.
