This book is a monograph about analytical and semi-analytical dynamics of nonlinear rotors. The analytical and semi-analytical solutions of periodic motion in nonlinear rotors are discussed. The generalized harmonic balance method for periodic motions to chaos in polynomial nonlinear systems is reviewed first and the semi-analytical method for periodic motions is reviewed second, which is also called as the implicit mapping method. The semi-analytical method can be used to any types of nonlinear dynamical systems. Analytical solutions for period-m motions in a buckled, nonlinear Jeffcott rotor system are obtained. The analytical expressions of periodic solutions are developed. The corresponding stability and bifurcation analyses of period-m motions are completed. A bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. A more accurate model of oil film forces is derived from Reynolds equations. The mechanical models of 2-DOF and 4-DOF nonlinear journal bearing rotor systems are established for a lower level amplitude transverse vibration. Such studies are to help one understand the nonlinear dynamics of rotor systems.