The M-estimate of a linear observation model has many important engineering applications such as identifying a linear system under non-Gaussian noise. Batch algorithms based on the EM algorithm or the iterative reweighted least squares algorithm have been widely adopted. In recent years, several sequential algorithms have been proposed. In this paper, we propose a family of sequential algorithms based on the Bayesian formulation of the problem. The basic idea is that in each step we use a Gaussian approximation for the posterior and a quadratic approximation for the log-likelihood function. The maximum a posteriori (MAP) estimation leads naturally to algorithms similar to the recursive least squares (RLSs) algorithm. We discuss the quality of the estimate, issues related to the initialization and estimation of parameters, and robustness of the proposed algorithm. We then develop LMS-type algorithms by replacing the covariance matrix with a scaled identity matrix under the constraint that the determinant of the covariance matrix is preserved. We have proposed two LMS-type algorithms which are effective and low-cost replacement of RLS-type of algorithms working under Gaussian and impulsive noise, respectively. Numerical examples show that the performance of the proposed algorithms are very competitive to that of other recently published algorithms.