This short volume reproduces in its first five chapters the essential content of a one semester course I gave at the University of Pennsylvania in the Spring of 1968) The last chapter covers material I had hoped to include, but could not for lack of time. My aim in giving such a course was to give students a body of material upon which some of the modern research in Diophantine geometry and higher arithmetic is based and to do this in a way which emphasized the many interesting roads out of these elementary foundations$ I wanted to enable them to start reading the literature in those areas of mathematics I personally find very interesting. The enthusiastic reception of my able class convinced me that this material was deserving of a larger audience, and I resolved to write up the notes for the course.