This book provides a valuable glimpse into discrete curvature, a rich new field of research that blends discrete mathematics, differential geometry, probability, and computer graphics. It includes a vast collection of ideas and tools that will offer something new to all interested readers. Discrete geometry has arisen as much from 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, and more. In spite of its crucial importance both in theoretical mathematics and in applications, almost no books have provided a coherent outlook on this emerging field.