The Shaggy Steed is an unassuming figure from Irish folklore who reveals himself as an inspiring teacher of the forces hidden in the universe. This book celebrates an unassuming bit of physics that also turns out to be an inspiring teacher. The two-body problem - the motion of two bodies bound by the inverse-square force of gravity and electricity - is the Shaggy Steed of physics, guiding the reader to an understanding of both the forces and the mathematical beauty hidden in the physical world. The book begins with an exposition of the action principle and its revelation of invariants created by the symmetries of nature. It then turns to the two-body problem and the grand unifying themes of symmetry and topology in physics, both classical and quantum. On the scale of the solar system this motion generates the Kepler ellipse - the fundamental orbit of celestial mechanics. On the microscopic scale of the quantum, the same motion generates the hydrogen atom - the primal element. This remarkable unity of the heavens and the elements rests upon hidden symmetry - the symmetry of rotations in four dimensions. This richly endowed symmetry also contains the symmetry of the relativistic space-time of Einstein. In addition it is the paradigm for the unitary symmetries of the elementary particles and pointed the way to them. This is a book for those seeking to know what breathes life into the equations of physics. It is also a fine tutorial book from which one can learn classical and quantum mechanics with pleasure. The Shaggy Steed of Physics is a spirited companion to the textbooks, a book that all students of physics, chemistry, and mathematics will want to read.