The investigation of nonlinear dynamis in physical and engineering systems from the point of view of systems and control theory is important to develop better engineering systems. Synchronization of oscillators and output regulation for rigid body systems are two problem classes which are inherently nonlinear and are of great importance in applications. This thesis contains novel results for both problem classes. In the case of sychronization of oscillators we consider two different system classes and give sufficient or necessary conditions for synchronization. In the case of the output regulation problems for rigid body systems we provide a new two-step control design procedure, a detailed analysis for the error dynamics and an application scenario for satellite control. A highlight of the thesis is a new separation principle which is the underlying principle of the two-step design procedure for the output regulation problem.