This work addresses the design of resilient estimators for stochastic systems. To this end, we introduce a minimum mean-squared error resilient (MMSE-R) estimator whose conditional mean squared error from the state remains finitely bounded and is independent of additive measurement attacks. An implementation of the MMSE-R estimator is presented and is shown as the solution of a semidefinite programming problem, which can be implemented efficiently using convex optimization techniques. The MMSE-R strategy is evaluated against other competing strategies representing other estimation approaches in the presence of small and large measurement attacks. The results indicate that the MMSE-R estimator significantly outperforms (in terms of mean-squared error) other realizable resilient (and non-resilient) estimators.