The reliable estimation of signals sent over noisy and lossy channels is a fundamental problem of communications and signal processing. This talk presents a general linear systems approach to analyzing and designing for such losses. The talk gives a new, general analysis of state estimation in “jump linear systems,” which are linear state space systems with discrete Markov dynamics. For communication problems, the discrete dynamics can model discrete changes in channel conditions such as sample erasures, while the continuous dynamics can model correlations in the source data to be transmitted. The main result provides a bound on the minimum achievable estimation error for a general jump linear system. A simple estimator that achieves this bound is also found and can be computed efficiently as a convex optimization with linear matrix inequality (LMI) constraints. A direct application of this framework provides bounds for tracking a time-varying signal from noisy observations with losses. Such bounds are relevant in designing sensor networks, where the communication of data may be subject to losses. Another application is a method for making standard predictive quantization robust to communication losses without channel coding.