We consider the problem of identifying multivariable finite dimensional linear time-invariant systems from noisy input/output measurements. Apart from the fact that both the measured input and output are corrupted by additive white noise, the output may also be contaminated by a term which is caused by a white input process noise; furthermore, all these noise processes are allowed to be correlated with each other. We shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open- or closed-loop. Two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms.