A modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasizing harmonic analysis on topological groups. The main goal is to cover John Tate's visionary thesis, providing virtually all of the necessary analytic details and topological preliminaries—technical prerequisites that are often foreign to the typical, more algebraically inclined number theorist. While most existing treatments of Tate's thesis are somewhat terse and less than complete, the intent here is to be more leisurely, more comprehensive, and more comprehensible. While the choice of objects and methods is naturally guided by specific mathematical goals, the approach is by no means narrow. In fact, the subject matter at hand is relevant not only to budding number theorists but also to students of harmonic analysis or the representation theory of Lie groups.