Omori-Yau maximum principle is a generalization of classic pointwise maximum principle for C2 smooth functions on compact manifolds to complete noncompact Riemannian manifolds with lower bound of Ricci curvature. It was introduced by S. T. Yau related to Liouville type theorem for harmonic functions on complete noncompact Riemannian manifolds with lower bound of Ricci curvature. It has been turned out to be useful in some geometric analysis problems on Riemannian manifolds. We consider its extension to Alexandrov spaces which appear as limit spaces of Riemannian manifolds under Gromov-Hausdorff distance.