Filtering methods are powerful tools to estimate the hidden state of a state-space model from observations available in real time. However, they are known to be highly sensitive to the presence of small misspecifications of the underlying model and to outliers in the observation process. In this article, we show that the methodology of robust statistics can be adapted to sequential filtering. We define a filter as being robust if the relative error in the state distribution caused by misspecifications is uniformly bounded by a linear function of the perturbation size. Since standard filters are nonrobust even in the simplest cases, we propose robustified filters which provide accurate state inference in the presence of model misspecifications. The robust particle filter naturally mitigates the degeneracy problems that plague the bootstrap particle filler (Gordon, Salmond, and Smith) and its many extensions. We illustrate the good properties of robust filters in linear and nonlinear state-space examples. Supplementary materials for this article are available online.