This article presents a complete review of the modeling and control schemes for overhead cranes operating in 2D and 3D spaces published to date. The modeling schemes including the pendulum-like models with rigid and flexible links are reviewed and their key characteristics are studied. Subsequently, an overview of the control methods developed for such models is presented. Afterward, a new simulation-oriented model enabling to capture both cables' dynamic and global nonlinearities caused by the pendulation is developed, and different control methods that exist in the literature are evaluated and compared based on this model using numerical experiments. In the end, several research gaps are identified to be considered in future works.