The theory of compressed sensing promises to revolutionize remote sensing, biomedical imaging and perhaps even digital photography. Mathematically, this theory is appealing because of the interplay of elements from random matrix theory, optimization theory and analysis. However, the randomized sensing model and the grid-based sparsity assumption central to many results are somewhat disconnected from typical signal spaces and sensor architectures used in engineering. This talk explores recent trends in narrowing the gap between theory and practice. Instead of sparsity in an orthonormal basis, we define a notion of sparsity in reproducing kernel spaces. The signal space is permitted to be infinite-dimensional while obtaining recovery from a finite number of measured quantities. The recovery procedure is based on optimization of a total variation norm. This work in collaboration with Gitta Kutyniok and Axel Flinth extends results by Candes and Fernandez-Granda as well as Recht et al.