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The use of multiple-antenna arrays can dramatically increase the throughput of wireless communication systems. Thus, it is important to characterize the statistics of the mutual information for realistic correlated channels. Here, a mathematical approach is presented, using the method of replicas, that provides analytic expressions not only for the average, but also for the higher moments of the distribution of the mutual information for the most general zero-mean Gaussian multiple-input multiple-output (MIMO) channels when the channel is known at the receiver. These channels include multitap delay paths, and channels with covariance matrices that cannot be written as a Kronecker product, such as general dual-polarized correlated antenna arrays. This approach is formally valid for large antenna numbers, in which case all cumulant moments of the distribution, other than the first two, scale to zero. In addition, it is shown that the replica-symmetric result is valid if the variance of the mutual information is positive and finite. In this case, it is shown that the distribution of the mutual information tends to a Gaussian, which enables the calculation of the outage capacity. These results are quite accurate even for few antennas, which makes this approach applicable to realistic situations.