One of the most important parts of engineered and biological systems is the ability to acquire and interpret information from the surrounding world accurately and in time-scales relevant to the tasks critical to system performance. This classical concept of efficient signal acquisition has been a cornerstone of signal processing research, spawning traditional sampling theorems (e.g. Shannon-Nyquist sampling), efficient filter designs (e.g. the Parks-McClellan algorithm), novel VLSI chipsets for embedded systems, and optimal tracking algorithms (e.g. Kalman filtering). Traditional techniques have made minimal assumptions on the actual signals that were being measured and interpreted, essentially only assuming a limited bandwidth. While these assumptions have provided the foundational works in signal processing, recently the ability to collect and analyze large datasets have allowed researchers to see that many important signal classes have much more regularity than having finite bandwidth. One of the major advances of modern signal processing is to greatly improve on classical signal processing results by leveraging more specific signal statistics. By assuming even very broad classes of signals, signal acquisition and recovery can be greatly improved in regimes where classical techniques are extremely pessimistic. One of the most successful signal assumptions that has gained popularity in recet hears is notion of sparsity. Under the sparsity assumption, the signal is assumed to be composed of a small number of atomic signals from a potentially large dictionary. This limit in the underlying degrees of freedom (the number of atoms used) as opposed to the ambient dimension of the signal has allowed for improved signal acquisition, in particular when the number of measurements is severely limited. While techniques for leveraging sparsity have been explored extensively in many contexts, typically works in this regime concentrate on exploring static measurement systems which result in static measurements of static signals. Many systems, however, have non-trivial dynamic components, either in the measurement system's operation or in the nature of the signal being observed. Due to the promising prior work leveraging sparsity for signal acquisition and the large number of dynamical systems and signals in many important applications, it is critical to understand whether sparsity assumptions are compatible with dynamical systems. Therefore, this work seeks to understand how dynamics and sparsity can be used jointly in various aspects of signal measurement and inference. Specifically, this work looks at three different ways that dynamical systems and sparsity assumptions can interact. In terms of measurement systems, we analyze a dynamical neural network that accumulates signal information over time. We prove a series of bounds on the length of the input signal that drives the network that can be recovered from the values at the network nodes~[1--9]. We also analyze sparse signals that are generated via a dynamical system (i.e. a series of correlated, temporally ordered, sparse signals). For this class of signals, we present a series of inference algorithms that leverage both dynamics and sparsity information, improving the potential for signal recovery in a host of applications~[10--19]. As an extension of dynamical filtering, we show how these dynamic filtering ideas can be expanded to the broader class of spatially correlated signals. Specifically, explore how sparsity and spatial correlations can improve inference of material distributions and spectral super-resolution in hyperspectral imagery~[20--25]. Finally, we analyze dynamical systems that perform optimization routines for sparsity-based inference. We analyze a networked system driven by a continuous-time differential equation and show that such a system is capable of recovering a large variety of different sparse signal classes~[26--30].