This text acquaints the reader with the essential ideas of analytic K-homology and develops some of its applications. It includes a detailed introduction to the necessary functional analysis, followed by an exploration of the connections between K-homology and operator theory, coarse geometry, index theory, and assembly maps, including a detailed treatment of the Atiyah-Singer Index Theorem. Beginning with the rudiments of C*-algebra theory, it leads the reader into some of the central notions of contemporary research in geometric functional analysis.