This book provides a comprehensive exploration in the fundamental physics of the non-relativistic atom in terms of its symmetries. The book gives a complete treatment of the dynamical symmetries of the H atom and the use of these symmetries to determine energy levels and analytically calculate the first order radiative Lamb shift using group theory. Understanding radiative shifts is critical since measurement of the Lamb shift led to the development of quantum electrodynamics. The harmony between the mathematics and the physics is emphasised throughout the book. Through the lens of symmetry, it is possible to elucidate key features of the H atom in a way that is elegant, simple, and focuses on fundamentals.Understanding the hydrogen atom has been the gateway to the development of quantum physics for over 100 years. Both quantum mechanics and quantum electrodynamics were discovered by exploring the spectra of the H atom. This book outlines the key mathematical and physical behavior of the H atom, and its historical context, that led to these discoveries. The quantum behavior is related to the corresponding classical behavior and to the harmonic oscillator, the other fundamental central force system. Understanding the H atom, the most basic physical system, is a gateway to understanding much of quantum theory.