This paper presents variable-and fixed-switching-frequency-based hysteresis current control (HCC) methods for single-phase grid-connected voltage-source inverters with LCL filter. The main feature of the proposed HCC methods is that the reference inverter current is generated through a proportional-resonant (PR) controller for achieving zero steady-state error in the grid current. The consequence of using the PR controller is eliminating the need for using derivative operations in generating the reference inverter current. Furthermore, the active damping method is employed to damp the LCL resonance. An equation is derived for variable-switching frequency. Fixed-switching frequency operation is achieved by modulating the hysteresis band. The performance of both HCC methods has been validated by simulation and experimentally. It is reported that the proposed HCC methods not only preserve the inherent features of the conventional HCC methods, but also damp the LCL resonance using an active damping method and guarantee zero steady-state error in the grid current.