Output only and input/output (I/O) system identification algorithms are developed based on a novel mean-square-error (MSE) criterion. The input is assumed non-Gaussian and the performance criterion implicitly exploits cumulant statistics to suppress the effect of additive Gaussian noise. The noise covariance need not be known, and in I/O problems both input and output (perhaps correlated) noises are allowed. Although expressed in terms of noisy data, the novel objective function is a scalar multiple of the standard MSE as if the latter was computed in the absence of noise. It yields strongly consistent parameter estimators which are obtained by solving linear equations via computationally attractive and noise insensitive recursive-least-squares and least-mean-squares algorithms. Simulations illustrate the performance of the proposed algorithms and they are compared them with the conventional methods.