These notes were written for a final year undergraduate course taught at Manchester University in 1988/9 and also taught in later years by Dr M. McCrudden. I rewrote them in 2000 to make them available to interested graduate students. The approach taken is very down to earth and makes few assumptions beyond standard undergraduate analysis and algebra. Because of this the course was as self contained as possible, covering basic number theory and analytic ideas which would probably be familiar to more advanced readers. The problem sets are based on those for produced for the course. I would Uke to thank Javier Diaz-Vargas and Jeremy Scofield for pointing out numerous errors.