Despite the inevitable presence of transceiver impairments, most prior work on multiple- input multiple-output (MIMO) wireless systems assumes perfect transceiver hardware, which is unrealistic in practice. In this direction, motivated by the increasing interest in MIMO relay systems due to their improved spectral efficiency and coverage, this paper investigates the impact of residual hardware impairments on the ergodic capacity of dual-hop (DH) amplify-and-forward (AF) MIMO relay systems. Specifically, a thorough characterization of the ergodic channel capacity of DH AF relay systems in the presence of hardware impairments is presented herein for both the finite and large antenna regimes by employing results from finite-dimensional and large random matrix theory, respectively. Regarding the former setting, we derive the exact ergodic capacity as well as closed-form expressions for tight upper and lower bounds. Furthermore, we provide an insightful study for the low signal-to-noise ratio regimes. Next, the application of the free probability theory allows us to study the effects of the hardware impairments in future 5G deployments including a large number of antennas. While these results are obtained for the large system limit, simulations show that the asymptotic results are quite precise even for conventional system dimensions.