The problem of filtering and smoothing for a system described by approximately linear dynamic and measurement relations has been studied for many decades. Yet the potential problem of misspecified dynamics, which makes the usual probabilistic assumptions involving normality and independence questionable at best, has not received the attention it merits. This paper proposes a probability-free multicriteria "flexible least squares" filter which meets this misspecification problem head on. A Fortran program implementation is provided for this filter, and references to simulation and empirical results are given. Although there are close connections with the standard Kalman filter, there are also important conceptual and computational distinctions. The Kalman filter, relying on probability assumptions for model discrepancy terms, provides a unique estimate for the state sequence. In contrast, the flexible least squares filter provides a family of state sequence estimates, each of which is vector-minimally incompatible with the prior dynamical and measurement specifications.