In this paper, we study the effect of channel estimation error at the receiver on the mutual information of a multiple-input-multiple-output (MIMO) channel obeying correlated Rayleigh fading. We assume that perfect knowledge of the channel correlation is available at the receiver and find upper and lower bounds on the mutual information for Gaussian input signals. We prove that for a generic input covariance matrix, the gap between these two bounds at high transmit powers does not exceed a certain value, which depends on the number of nonzero eigenvalues of the receive correlation matrix. We also show that in a system with uncorrelated transmit antennas and correlated receive antennas and with uniform power distribution over the transmit antennas, these bounds are asymptotically tight for a large number of transmit antennas. We show that when the correlation information is available at the transmitter, the transmitting strategy that maximizes the mutual information lower bound is to transmit toward the directions of the eigenvectors of the transmit correlation matrix. We further derive upper and lower bounds on the mutual information of a MIMO multiple-access channel with imperfect channel estimation at the receiver. We also prove that when the input power at each user is uniformly distributed over its transmit antennas, the bounds on the mutual information are asymptotically tight for Gaussian input signals, and this tightness increases when the number of users increases. Numerical simulations are conducted to corroborate theoretical results.