State estimation is a crucial part of the monitoring and/or control of all chemical processes. Among various approaches for this problem, moving horizon estimation (MHE) has the advantage of directly incorporating nonlinear dynamic models within a well-defined optimization problem. Moreover, advanced step moving horizon estimation (asMHE) substantially reduces the on-line computational expense associated with MHE. Previously, MHE and asMHE have both been shown to perform well when measurement noise follows some known Gaussian distribution. In this study we extend MHE and asMHE to consider measurements that are contaminated with large errors. Here standard least squares based estimators generate biased estimates even with relatively few gross error measurements. We therefore apply two robust M-estimators, Huber's fair function and Hampel's redescending estimator, in order to mitigate the bias of these gross errors on our state estimates. This approach is demonstrated on dynamic models of a CSTR and a distillation column. Based on this comparison we conclude that the asMHE formulation with the redescending estimator can be used to get fast and accurate state estimates, even in the presence of many gross measurement errors. (C) 2013 Elsevier Ltd. All rights reserved.