This paper provides the analysis of capacity expressions for multi-user and multi-cell wireless communication schemes when the transmitters and the receivers have a large number of correlated antennas. Our main contribution mathematically translates into a deterministic equivalent of the Shannon transform of a class of large dimensional random matrices with possibly asymptotically large eigenvalues. On the applicative side, this class of large matrices is used in this contribution to model (i) multi-antenna MAC and BC with transmit and receive channel correlation, (ii) MIMO communications with inter-cell interference and channel correlation both at the base stations and at the receivers. The theoretical capacity formulas obtained for these models extend the classical results on multi-user MIMO capacities in independent and identically distributed Gaussian channels to the more realistic Gaussian channels with separable variance profile. On an information theoretical viewpoint, this article provides: in scenario (i), an asymptotic description of the MAC and BC rate regions as a function only of the transmit and receive correlation matrices (thus independently of the random channel realizations); an asymptotic expression of the capacity-maximizing covariance matrices in the uplink MAC and downlink BC; a water-filling algorithm which, upon convergence, is proved to converge to the capacity-achieving power allocation at the transmitters both in MAC and BC. In scenario (ii), the article provides: an expression of the single-user decoding capacity and MMSE decoding capacity when interference is treated as Gaussian noise with a known variance profile; for single-user decoding, the capacity-achieving antenna power allocation policy at the transmitter.