The purpose of this book is to introduce the reader to the arithmetic of quadratic forms. Quadratic forms in this book are mainly considered over the rational number field or the ring of rational integers and their completions. It is of course possible to discuss quadratic forms over more general number theoretic fields and rings. But it is not hard to understand the general theory once readers have learned the simpler theory in this book. This book is self-contained except for the following: The theory of quadratic fields is used to prove the first approximation theorem in Chapter 6. In Chapter 7, the relative theory is treated and so the use of algebraic number fields is inevitable. Dirichlet's theorem on prime numbers in arithmetic progressions is also used without proof.

Several exercises are given. The answers are known anyway. Some of them are easy and some of them are not. Some problems are also given in the Notes. Problems mean questions for which I do not know the answer and want to know it. They are very subjective and arbitrary. I do not say that they have citizenship in the world of quadratic forms.

Inheriting pioneering works of Eisenstein, Smith, Minkowski, Hardy and others, Hasse established the so-called Minkowski-Hasse principle on quadratic forms and Siegel gave the so-called Siegel formula and other great works. Some of them were formulated from the view point of algebraic groups, and stimulated and developed them. One of the achievements of Eichler was to introduce the important concept of spinor genus. Kneser showed that group-theoretic consideration was fruitful. Their method gave a unified perspective to sporadic results.