Volumetric PolyCube-Map-based methods off er automatic ways to construct all-hexahedral meshes for closed 3D polyhedral domains, but their meshing quality is limited by the lack of interior singularities and feature alignment. In the presented work, we propose cut-enhanced PolyCube-Maps, to introduce essential interior singularities and preserve most input features. Our main idea is simple and intuitive: by inserting proper parameterization seams into the initial PolyCube-Map via novel PolyCube cutting operations, the mapping distortion can be reduced significantly.The cut-enhanced PolyCube-Map computation includes feature-aware PolyCube-Map construction and cut-enhanced PolyCube deformation. The former aims to preserve input feature edges during the initial PolyCube-Map construction. The latter introduces seams into the volumetric PolyCube shape by cutting it through selective PolyCube edges and deforms the modified PolyCube under the seamless constraints to compute a low-distortion PolyCube-Map. The hexahedral mesh induced by the fi nal PolyCube-Map can be further enhanced by our mesh improvement algorithm. We validate the efficacy of our method on a collection of more than one hundred CAD models and demonstrate its advantages over other automatic all-hex meshing methods and padding strategies. The limitations of cut-enhanced PolyCube-Maps are also discussed thoroughly.