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In this paper we mainly focus on the information-theoretic capacity of multiple-input multiple-output (MIMO) independent and identically distributed (i.i.d.) Rayleigh flat-fading channels assuming equal power allocation to each of the transmit antennas. We present an overview of the analytical results on MIMO channel capacity in recent years, including the statistical properties, approximate distributions and lower and upper bounds of ergodic capacity and so on. These simple analytical results can enable us to evaluate and optimize MIMO systems efficiently. Firstly, exact mean and variance of MIMO capacity are discussed based on random matrix theory and the asymptotic results are given in closed-form formulas when the numbers of transmit and receive antennas get large at constant ratio. Then, Gaussian and Gamma distributions are proposed to approximate the distribution of MIMO capacity with the known capacity mean and variance. Afterwards, some lower and upper bounds of ergodic capacity are presented with the aid of matrix inequality theory. Finally, the accuracy of Gaussian and Gamma approximations and the tightness of ergodic capacity bounds are investigated in comparison with Monte Carlo (MC) results. The numerical simulations validate the effectiveness of the analytical results.