The subject of mesh generation and optimization is very important in many scientific applications. In this paper, we investigate the issue of mesh optimization via the construction of Centroidal Voronoi Tessellations. Given some initial Delaunay meshes with only average quality, it is shown that the CVT based mesh optimization generates a robust, high quality mesh which does not rely critically on the choice of the initial mesh. In comparison, other smoothing techniques, such as the classical Laplacian smoothing, tend to be more sensitive to the initial distributions of vertices. Thus, the CVT based optimization may be advocated as a preferred choice for mesh optimization and smoothing.