This book is a monograph about the analytical dynamics of nonlinear rotor systems. The analytical solutions of periodic motions in nonlinear rotor systems are presented in this book. To help one understand the analytical solutions, the generalized harmonic balance method for periodic motions in polynomial nonlinear systems is briefly reviewed first, and then the semi-analytical method for periodic motions in any nonlinear systems is presented briefly too, which is also called the implicit mapping method. Such two analytical methods are employed to determine the solutions of periodic motions in nonlinear rotor systems. The analytical expressions of periodic motions to chaos for nonlinear rotor systems are presented, and the frequency-amplitude characteristics of nonlinear rotor systems are discussed. In addition, the accurate modeling of nonlinear rotors with oil films is presented, and the periodic motions of such fully nonlinear oil-film rotor systems are developed through the semi-analytical method. This book provides a better understanding of frequency-amplitude characteristics in nonlinear rotor systems. The methodology presented in this book can help one study complicated nonlinear rotor systems.