Delaunay triangulations and Voronoi diagrams are important tools in many fields like Astronomy, Physics, Chemistry, Biology, Ecology, Economics, Mathematics and Computer Science. They are oftentimes used in problems related to the generation meshes. The main purpose of this book is to study the interactions between these two geometrical structures and sharp approximations using enrichment functions. This book provides some old and some new results, concepts, and further developments in the area of high-dimensional approximation of functions or their integrals and their applications in finite element methods, which are based upon a geometric point of view exploiting Delaunay triangulations and Voronoi tessellations or some of their generalizations. This book is of particular concern to the scientific community in general and, in particular, to young researchers preparing a Ph.D. in analysis, but also postgraduates and more advanced researchers, who are interested in the areas of mathematical analysis with a major concentration in Numerical Analysis (Applied Mathematics). Specifically, those working in the theory of functions, enriched finite-element approximation, geometric function theory, and optimization will find new insights as well as a guide to advanced topics. It would also assist as an invaluable reference text for the experts. The novelty of this work lies in his profound abilities and skills in applying techniques from other areas of mathematics such as approximation theory and optimization theory, to obtain final answers to countless open problems. This book gives relatively complete proofs of theorems from many different fields of mathematics, and often gives multiple proofs for the same theorem, followed by non-trivial (and sometimes new) examples. In addition, this book also presents interesting, important unsolved problems in enriched finite element approximation.