This paper proposes a novel scheme for the current controller for the grid-side converter (GSC) of permanent-magnet synchronous generator (PMSG) wind turbines to eliminate the high-order harmonics in the grid currents under grid voltage disturbances. The voltage unbalance and harmonics in three-phase systems cause grid current distortions. In order to mitigate the input current distortions, multi-loop current controllers are applied, where the positive-sequence component is regulated by proportional-integral (PI) controllers, and the negative-sequence and high-order harmonic components are regulated by proportional-resonance (PR) controllers. For extracting the positive/negative-sequence and harmonic components of the grid voltages and currents without a phase delay or magnitude reduction, composite observers are applied, which give faster and more precise estimation results. In addition, an active damping method using PR controllers to damp the grid current component of the resonant frequency is employed to improve the operating stability of VSCs with inductor-capacitor-inductor (LCL) filters. The validity of the proposed method is verified by simulation and experimental results.