The mixed radio frequency (RF)/free-space optical (FSO) relaying is a promising technology for coverage improvement, while there lacks unified expressions to describe its performance. In this paper, a unified performance analysis framework of a dual-hop relay system over asymmetric RF/FSO links is presented. More specifically, we consider the RF link follows generalized $κ$-$μ$ or $η$-$μ$ distributions, while the FSO link experiences the gamma-gamma distribution, respectively. Novel analytical expressions of the probability density function and cumulative distribution function are derived. We then capitalize on these results to provide new exact analytical expressions of the outage probability and bit error rate (BER). Furthermore, the outage probability for high signal-to-noise ratios and the BER for different modulation schemes are deduced to provide useful insights into the impact of system and channel parameters of the overall system performance. These accurate expressions are general, since they correspond to generalized fading in the RF link and account for pointing errors, atmospheric turbulence and different modulation schemes in the FSO link. The links between derived results and previous results are presented. Finally, numerical and Monte-Carlo simulation results are provided to demonstrate the validity of the proposed unified expressions.