Tracking time-varying signals is an important part of forecasting in complex time-series data. Recently, signal processing techniques have been developed to improve tracking performance when the signal of interest is known a-priori to be sparse. In this talk we will review a collection of related algorithms we have developed for dynamic filtering of time-varying sparse signals. The foundations of this work are based on two algorithms that leverage efficient L1 optimization methods. The first is a simple algorithm (BPDN-DF) that works well when the system dynamics are known exactly and where analytic guarantees are available. The second is a novel algorithm (RWL1-DF) that is more computationally complex than BPDN-DF but performs better in practice, especially in the case where the system dynamics model is inaccurate. Robustness to model inaccuracy is achieved by using a hierarchical probabilistic data model and propagating higher-order statistics from the previous estimate (akin to Kalman filtering) in the sparse inference process. Following this, we will examine extensions of these basic algorithms for specialized circumstances such as earth mover's distance regularization for ordered spaces and sparse bayesian learning adaptations for inverse problems with high coherence. We will demonstrate these algorithms on problems from image processing, computational imaging and neurophysiology data analysis.