At the time that the first edition was written, there had been relatively little recent work on the central theory of the Hardy-Littlewood method, namely that surrounding Waring's problem and associated questions. Indeed, the work of Davenport and Vinogradov had taken on the aspect of being written on tablets of stone. This is in complete contrast to the current situation. In the last decade or so there has been a series of important developments in the area. The tract is, therefore, ripe for revision, and the opportunity has been taken to give an introduction to this new material, and especially to the important work of Wooley. Chapter 5 has been extensively rewritten to take account of our new understanding of Vinogradov's mean value theorem, and a completely new chapter has been added to describe the new work on Waring's problem. Fortunately the large bulk of the material has not been superseded and the underlying ideas still play an important rôle in many of the new developments.