The paper discusses the properties and applications of centroidal Voronoi tessellations, which are Voronoi tessellations where the generating points are the centroids of the corresponding Voronoi regions. The authors provide applications in image compression, quadrature, finite difference methods, resource distribution, cellular biology, statistics, and animal territorial behavior. They also present methods for computing these tessellations, analyze their properties, and report numerical experiments. The paper covers various metrics and generalizations of Voronoi regions, and discusses the minimization of functional errors in quadrature rules and optimal representation problems. It highlights the connection between centroidal Voronoi diagrams and optimal clustering, and provides examples of their use in finite difference schemes and resource placement. The authors also explore the application of centroidal Voronoi tessellations in non-Euclidean metrics and discuss the existence and uniqueness of minimizers for these tessellations.The paper discusses the properties and applications of centroidal Voronoi tessellations, which are Voronoi tessellations where the generating points are the centroids of the corresponding Voronoi regions. The authors provide applications in image compression, quadrature, finite difference methods, resource distribution, cellular biology, statistics, and animal territorial behavior. They also present methods for computing these tessellations, analyze their properties, and report numerical experiments. The paper covers various metrics and generalizations of Voronoi regions, and discusses the minimization of functional errors in quadrature rules and optimal representation problems. It highlights the connection between centroidal Voronoi diagrams and optimal clustering, and provides examples of their use in finite difference schemes and resource placement. The authors also explore the application of centroidal Voronoi tessellations in non-Euclidean metrics and discuss the existence and uniqueness of minimizers for these tessellations.