This paper proposes a unified statistical channel model to characterize the atmospheric turbulence induced distortions faced by orbital angular momentum (OAM) in free space optical (FSO) communication systems. In this channel model, the self-channel irradiance of OAM modes as well as crosstalk irradiances between different OAM modes are characterized by a Generalized Gamma distribution (GGD). The latter distribution is shown to provide an excellent match with simulated data for all regimes of atmospheric turbulence. Therefore, it can be used to overcome the computationally complex numerical simulations to model the propagation of OAM modes through atmospheric turbulent FSO channels. The GGD allows obtaining very simple tractable closed-form expressions for a variety of performance metrics. Indeed, the average capacity, the bit-error rate, and the outage probability are derived for FSO systems using single OAM mode transmission with direct detection. Furthermore, we extend our study to FSO systems using OAM mode diversity to improve the performance. By using a maximum ratio combining (MRC) at the receiver, the GGD is also shown to fit the simulated combined received optical powers. Finally, space-time (ST) coding is proposed to provide diversity and multiplexing gains, and the error probability is theoretically derived under the newly proposed generic model.