The fundamental connection between the stability of linear filtering and linear systems theory was already remarked in Kalman’s seminal 1960 paper. Unfortunately, the linear theory relies heavily on the investigation of the explicit Kalman filtering equations, and sheds little light on the behavior of nonlinear filters. Nonetheless, it is possible to establish surprisingly general connections between the stability of nonlinear filters and nonlinear counterparts of basic concepts in linear systems theory: stability, observability, detectability. The proofs of these results are probabilistic in nature and provide significant insight into the mechanisms that give rise to filter stability. The aim of this paper is to review these recent results and to discuss some of their applications.