This book provides a valuable glimpse into discrete curvature, a rich new field of research which blends discrete mathematics,
differential geometry, probability and computer graphics. It includes a vast collection of ideas and tools which will offer
something new to all interested readers. Discrete geometry has arisen as much as a theoretical development as in response
to unforeseen challenges coming from applications. Discrete and continuous geometries have turned out to be intimately connected.
Discrete curvature is the key concept connecting them through many bridges in numerous fields: metric spaces, Riemannian and
Euclidean geometries, geometric measure theory, topology, partial differential equations, calculus of variations, gradient
flows, asymptotic analysis, probability, harmonic analysis, graph theory, etc. In spite of its crucial importance both in
theoretical mathematics and in applications, up to now, almost no books have provided a coherent outlook on this emerging