Many problems in science and engineering require the reconstruction of an object - an image or signal, for example - from a collection of measurements. Due to time, cost or other constraints, one is often severely limited by the amount of data that can be collected. Compressed sensing is a mathematical theory and set of practical techniques that aim to improve reconstruction quality from seemingly incomplete data sets by exploiting the underlying structure of the unknown object; specifically, its sparsity. In this talk I will commence with an overview of the fundamentals of compressed sensing and discuss some of its applications. However, I will next show that, despite the large and growing body of literature on compressed sensing, many of these applications do not fit into the existing theory. I will then describe a more general framework for compressed sensing that bridges this gap. Finally, I will demonstrate that this framework is not just useful in explaining existing applications of compressed sensing, but that the new insight it brings leads to substantially better compressed sensing approaches than the current state-of-the-art.