In 1982, I was invited to give a course of 11 two-hour lectures in the University of Nagoya on some branch of Commutative Algebra. The topic I chose was the asymptotic theory of ideals and the lectures were duly given between December 1982 and March 1983. The notes below are an extensive revision of the notes given to the audience at the lectures and, with certain exceptions, the chapter headings below correspond to the titles of the individual lectures. The exceptions referred to are the following. First, the notes of the third lecture have been considerably expanded so as to incorporate a proof of the Mori-Nagata Theorem, based on the beautiful theorem of Matijevic, and the original topic of the third lecture, the Valuation Theorem, is dealt with in the fourth lecture. The second change is more considerable. The last three lectures of the course dealt with Teissier's theory of mixed multiplicities as given in Teissier[1973] and was based on the use of complete and joint reductions of a set of ideals. In the last lecture I applied these ideas to prove what I call the general degree formula. An account of the theory of complete and joint reductions has since appeared in Rees[1984], while, since the lectures were given, I have succeeded in proving a still more general degree formula using a quite different method.