Many classical and modern results about quadratic forms are brought together in this book. The treatment is self-contained and of a totally elementary nature requiring only a basic knowledge of rings, fields, polynomials and matrices in order to be able to tackle the work of Pfister, Hilbert, Radon, Hurwitz, Pourchet and others as it relates to the study of numbers that can be expressed as squares, or sums of squares. The author deals with different approaches to their study, from classical results to the area of current research. This will be a fascinating volume for mathematicians in number theory or algebra.