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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 converges in distribution to a Gaussian random variable, and give closed-form formulas for its mean and variance. These results enable us to derive the first asymptotic formula for outage rates, 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 simulation that they are often quite accurate, even for relatively small arrays.