This book provides the very first comprehensive and self-contained introduction to hedgehog theory, which is born of the desire to visualize the formal differences of convex bodies. This extension of convex geometry has revealed unexpected depth and connections with many areas of mathematics, shedding new light on old problems such as the characterization of the 2-sphere conjectured by A.D. Alexandrov in the 1930s. The author is particularly keen to demonstrate the breadth and variety of applications of hedgehogs and their generalizations, in both geometry and analysis. Researchers in convex or differential geometry, as well as specialists in Monge-Ampère PDEs, will certainly find it a source of inspiration.