The objective of this study is to design a new anti-swing control scheme for overhead cranes, the performance of which equals or possibly surpasses the performance of a skilled crane operator. In this study, the anti-swing control problem is solved as a trajectory-tracking control problem. An anti-swing trajectory control scheme is designed based on the Lyapunov stability theorem, and then extended to an adaptive scheme to cope with parametric uncertainties, in which sufficient damping of load swing is achieved by modifying the reference trajectory of the trolley. The proposed control consists of a feed forward control and a non-linear PID control with gravity compensation, which guarantees asymptotic stability while keeping all internal signals bounded. The proposed control, allowing high-speed load hoisting, also guarantees accurate control of trolley position and rope length with optimum damping. In summary, the proposed control realizes a typical crane operation in practice, maintains its control performance in the presence of parametric uncertainties, and has a simple structure for easy implementation and gain tuning. The performance of the proposed control is shown with computer simulations.