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This tract gives a fairly elementary account of the theory of quadratic forms with integral coefficients and variables. It assumes a knowledge of the rudiments of matrix algebra and of elementary number theory, but scarcely any analysis. It is therefore intelligible to beginners and helps to prepare them for the study of the advanced work on quadratic forms over general rings. Dr Watson works step by step from wider (and easier) to narrower relations between forms, the final goal being the study of equivalence. The important problem of representation of integers is fully discussed in the course of the main development. There is an early chapter on reduction. Existing work on the theory of integral quadratic forms is obscure (partly for historical reasons). But the straightforward approach adopted by Dr Watson leads to a consideration of most of the main problems; there are proofs of many recent results, including some discovered by Dr Watson but hitherto unpublished.