Filtering methods are powerful tools to estimate the hidden state of a statespace 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 paper, we show that the methodology of robust statistics can be adapted to sequential filtering. We introduce an impact function that quantifies the sensitivity of the state distribution with respect to new data. Since the impact function of standard filters are unbounded even in the simplest cases, we propose filters with bounded impact functions which provide accurate state and parameter inference in the presence of model misspecifications. In particular, the robust particle filter naturally solves the degeneracy problems that plague the bootstrap particle filter (Gordon, Salmond and Smith, 1993) and its many extensions. We illustrate the good properties of robust filters in several examples, including linear state-space models and nonlinear models of stochastic volatility.