Spontaneous symmetry breaking is a revolutionary new idea of thinking in terms of symmetry and the corresponding properties of physical systems. The first part of this book develops the mathematical understanding of spontaneous symmetry breaking on the basis of classical field theory. The author addresses the existence of sectors, stability, and presents an improved Noether theorem, as well as the classical counterpart of the Goldstone theorem. The second part deals with quantum systems. Here criteria for spontaneous symmetry breaking are discussed in detail as well as Goldstone's theorem, gauge theories are included, and the theorem on the Higgs mechanism is presented. The author keeps the mathematical details to the minimum required to make the book accessible to students with basic knowledge of Hilbert space structures. Much of the material appears here for the first time in book form.