Abstract: This paper introduces L_p-Centroidal Voronoi Tessellation (L_p-CVT), a generalization of CVT that minimizes a higher-order moment of the coordinates on the Voronoi cells. This generalization allows for aligning the axes of the Voronoi cells with a predefined background tensor field (anisotropy). L_p-CVT is computed by a quasi-Newton optimization framework, based on closed-form derivations of the objective function and its gradient. The derivations are given for both surface meshing ($omega $ is a triangulated mesh with per-facet anisotropy) and volume meshing ($omega $ is the interior of a closed triangulated mesh with a 3D anisotropy field). Applications to anisotropic, quad-dominant surface remeshing and to hexdominant volume meshing are presented. Unlike previous work, L_p-CVT captures sharp features and intersections without requiring any pre-tagging.