This book thoroughly examines the nonlinear phenomena of the shedding in the wake and the fluid forces on a cylinder and how they depend on its oscillatory motion when they are synchronized (or locked) or nonsynchronized with the oscillation. The effect of the oscillation orientation and pattern is significant and result in very different synchronization modes. The book also discusses and develops reduced-order models based on nonlinear ordinary-differential equations or nonlinear algebraic equations, which produce the lift and drag coefficients on a cylinder or a cylinder-like structure. Computational fluid dynamics, analytical approaches, high-order spectral analysis, and modern techniques of nonlinear dynamics are used heavily as powerful tools throughout the book.