Generalized fading has been an imminent part and parcel of wireless communications. It not only characterizes the wireless channel appropriately but also allows its utilization for further performance analysis of various types of wireless communication systems. Under the umbrella of generalized fading channels, a unified ergodic capacity analysis of a free-space optical (FSO) link under both types of detection techniques (i.e., intensity modulation/direct detection (IM/DD) as well as heterodyne detection) over generalized atmospheric turbulence channels that account for generalized pointing errors is presented. Specifically, unified exact closed-form expressions for the moments of the end-to-end signal-to-noise ratio (SNR) of a single link FSO transmission system are presented. Subsequently, capitalizing on these unified statistics, unified exact closed-form expressions for ergodic capacity performance metric of FSO link transmission systems is offered. Additionally, for scenarios wherein the exact closed-form solution is not possible to obtain, some asymptotic results are derived in the high SNR regime. All the presented results are verified via computer-based Monte-Carlo simulations.