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Written by well-known expert Donald Saari, this book is directed toward readers who want to learn about the Newtonian $N$-body problem. It is also intended for students and experts who are interested in new expositions of past results in this area, previously unpublished research conclusions, and new research problems. Professor Saari has written the book for a broad audience, including readers with no previous knowledge about this fascinating area. He begins each chapter with introductory material motivated by unanswered research questions. He then includes some history, discussions intended to develop intuition, descriptions of open problems, and examples taken from real problems in astronomy.The first chapter starts with simple explanations of the apparent 'looping' orbit of Mars and the unexpected 'Sunrise, Sunset' behavior as viewed from Mercury and then leads up to the unexplained and weird dynamics exhibited by Saturn's F-ring. The second chapter, which introduces a way to decompose the velocity of the system, is motivated by a seemingly easy but unanswered conjecture involving the dynamics of the system when the system's diameter is a constant. The third chapter, which describes questions about the structure of the rings of Saturn, introduces new and surprisingly simple ways to find configurations of the particles that are 'central' to any discussion of the $N$-body problem. The fourth chapter analyzes collisions, and the last chapter discusses the likelihood of collisions and other events. The book is suitable for graduate students and researchers interested in celestial mechanics.