Variation et Optimisation de Formes is an introduction to the modern theory of shape optimization. There are no prerequisites beyond a standard undergraduate background in mathematics. Yet the book leads up to open questions in this very active field of research. Methodology and tools of mathematical analysis and geometry for variations of shapes are carefully described. Emphasis is put on geometric questions for the Laplace operator, on the classical capacity, or on shape derivatives. It also gives an FAQ on the usual topologies for domains and on the geometric properties of optimal shapes and shows what to do when they do not exist. The book ends on an extended bibliography.