This is the first monograph to exclusively treat Kac-Moody (K-M) groups, a standard tool in math and math physics, with connections to number theory, combinatorics, topology, singularities, quantum groups, and completely integrable systems. A self-contained text, systematically presented from the Lie algebra to the broader K-M Lie group setting. No prior knowledge of K-M Lie algebras is required. Exposition features numerous examples, illustrations, challenging problems, exercises, and is suitable for grad courses. Also of interest to researchers at the crossroads of representation theory, geometry, and topology.