Synopsis This book provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object. Shapes and geometries: analysis, Differential calculus, and optimization presents the extensive, recently developed theoretical foundation to shape optimization in a form that can be used by the engineering community. It also clearly explains the state-of-art developments in a mathematical language that will attract  mathematicians to open questions in this important field.