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Free-space optical (FSO) communication enables high rate data transmission over the atmospheric channel. However, turbulence-induced fading poses severe challenges for operating these systems. Employing spatial diversity has been proposed as an effective measure to improve system performance. In this paper, we first develop a comprehensive model for multiple-input multiple-output (MIMO) coherent and differential FSO systems with heterodyne detection taking into account all relevant signal and noise terms. For the sake of comparison, a similar model is also developed for direct detection. An asymptotic performance analysis is presented for MIMO FSO systems employing coherent and differential space-time codes (STCs) over Gamma-Gamma fading channels. We consider the practically important case of two transmit and an arbitrary number of receive apertures and provide a simple STC design criterion for coherent and differential FSO systems. Our results reveal that, in contrast to FSO systems with intensity modulation and direct detection (IM/DD), for coherent and differential FSO systems, orthogonal space-time block codes previously introduced in the RF literature, are preferable over repetition STCs, which do not yield full diversity. We quantify the asymptotic performance gain of coherent space-time coded FSO systems compared to IM/DD systems and show that this gain is mainly caused by the superiority of heterodyne detection compared to direct detection rather than the different STC and modulation designs.