Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations where the generating points are the centroids of their corresponding Voronoi regions. This paper discusses applications of CVTs in image compression, quadrature, finite difference methods, resource distribution, cellular biology, statistics, and animal territorial behavior. It also presents methods for computing CVTs, analyzes their properties, and reports numerical experiments. CVTs are particularly useful in data compression, where they help reduce the number of colors in an image while maintaining visual quality. The paper also explores the use of CVTs in optimal quadrature rules, where they provide accurate approximations of integrals. CVTs are used in clustering analysis, where they help partition data into clusters that best represent the data. The paper also discusses the use of CVTs in modeling cell division and territorial behavior in animals. The paper concludes with a discussion of the theoretical properties of CVTs and their computational methods.Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations where the generating points are the centroids of their corresponding Voronoi regions. This paper discusses applications of CVTs in image compression, quadrature, finite difference methods, resource distribution, cellular biology, statistics, and animal territorial behavior. It also presents methods for computing CVTs, analyzes their properties, and reports numerical experiments. CVTs are particularly useful in data compression, where they help reduce the number of colors in an image while maintaining visual quality. The paper also explores the use of CVTs in optimal quadrature rules, where they provide accurate approximations of integrals. CVTs are used in clustering analysis, where they help partition data into clusters that best represent the data. The paper also discusses the use of CVTs in modeling cell division and territorial behavior in animals. The paper concludes with a discussion of the theoretical properties of CVTs and their computational methods.