This book can be divided into two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. The raison d’ˆ etre of the book is in the second part, and so let us first explain the contents of the second part. There are two principal topics: (A) Classification of quadratic forms; (B) Quadratic Diophantine equations. Topic (A) can be further divided into two types of theories: (a1) Classification over an algebraic number field; (a2) Classification over the ring of algebraic integers. To classify a quadratic form— over an algebraic number field F, almost all previous authors followed the methods of Helmut Hasse. Namely, one first takes— in the diagonal form and associates an invariant to it at each prime spot of F, using the diagonal entries. A superior method was introduced by Martin Eichler in 1952, but strangely it was almost completely ignored, until I resurrected it in one of my recent papers. We associate an invariant to— at each prime spot, which is the same as Eichler’s, but we define it in a different and more direct way, using Clifford algebras. In Sections 27 and 28 we give an exposition of this theory. At some point we need the Hasse norm theorem for a quadratic extension of a number field, which is included in class field theory. We prove it when the base field is the rational number field to make the book self-contained in that case.