  We show that in a locally compact complete $CAT(0)$ space satisfying positive
angles property and a disintegration regularity for its canonical Hausdorff
measure, there exists a unique optimal transport map that push-forwards a given
absolutely continuous probability measure to another probability measure. In
particular this holds for the Riemannian manifolds of non-positive sectional
curvature and $CAT(0)$ Euclidean polyhedral complexes. Moveover we give a polar
factorization result for Borel maps in $CAT(0)$ spaces in terms of optimal
transport maps and measure preserving maps.
