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We consider the asymptotic behavior of the capacity of multiple antenna Rayleigh fading channels in the limit as the transmit and receive arrays become large. We show that the capacity random variable has a limiting Gaussian distribution, and give closed form formulas for its mean and variance when the arrays are uncorrelated. We also derive a more general formula for the variance in terms of the ergodic capacity which is applicable when either the transmit or the receive arrays are correlated. These results enable us to derive the first asymptotic formula for outage capacity, as well as a sharper estimate of the error in previously reported asymptotic formulas for ergodic capacity. Although these formulas are asymptotic, we show by simulations that they are often quite accurate, even for relatively small arrays.