We present several variations on Centroidal Voronoi Tesselations. First we review the classical definition, as a stable critical point of an objective function (quantization noise power), then we propose some modifications of the objective function (anisotropy, Lp norm). The so-modified Centroidal Voronoi Tesselations are useful for applications in geometry processing. Thus we demonstrate feature-aware surface remeshing, hexaedral-dominant meshing of 3D domains and fitting subdivision surfaces to unstructured triangle sets.