Gitta Kutyniok Technische Universität Berlin Program in Applied and Computational Mathematics Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements by using efficient algorithms. In this talk, we will first give an introduction to compressed sensing. We will then focus on sparsifying systems provided by applied harmonic analysis such as Gabor systems, wavelet systems, or shearlet systems, which typically provide provably optimal sparse approximations within a certain model situation. Combining these concepts, we will show that several problems such as feature detection can be effectively solved.