This is a sequel to my earlier book, A Concise Introduction to the Theory of Numbers. The latter was based on a short preparatory course of the kind traditionally taught in Cambridge at around the time of publication about 25 years ago. Clearly it was in need of updating, and it was originally intended that a second edition be produced. However, on looking through, it became apparent that the work would blend well with more advanced material arising from my lecture courses in Cambridge at a higher level, and it was decided accordingly that it would be more appropriate to produce a substantially new book. The now much expanded text covers elements of cryptography and primality testing. It also provides an account of number fields in the classical vein including properties of their units, ideals and ideal classes. In addition it covers various aspects of analytic number theory including studies of the Riemann zetafunction, the prime-number theorem, primes in arithmetical progressions and a brief exposition of the Hardy–Littlewood and sieve methods. Many worked examples are given and, as with the earlier volume, there are guides to further reading at the ends of the chapters.