Significantly non-normal noise, and particularly the presence of outliers, severely degrades the performance of the Kalman Filter, resulting in poor state estimates, non-white residuals, and invalid inference. An approach to robustifying the Kalman Filter based on minimax theory is described. The relationship between the minimax robust estimator of location formulated by Huber, its recursive versions based on the stochastic approximation procedure of Robbins and Monro, and an approximate conditional mean filter derived via asymptotic expansion, is shown. Consistency and asymptotic normality results are given for the stochastic approximation recursion in the case of multivariate time-varying stochastic linear dynamic systems with no process noise. A first-order approximation is given for the conditional prior distribution of the state in the presence of e-contaminated normal observation noise and normal process noise. This distribution is then used to derive a first-order approximation of the conditional mean estimator for the case where both observation and process noise are present.