The above heading is the title of a new book to be published by Cambridge University Press. In these lectures I will cover some of the main issues discussed in this monograph/textbook. In particular, we will discuss how the key to the success of compressed sensing applied in imaging lies in the structure. For example images are not just sparse in an X-let expansion, they have a very specific sparsity structure in levels according to the X-let scales. Similarly, when considering Total Variation, the gradient coefficients are also highly structured. Moreover, in most realistic sampling scenarios, the sampling operator combined with any X-let transform yields a matrix with a very specific coherence structure. The key to successfully use compressed sensing is therefore to understand how to utilise these structures in an optimal way, in particular in the sampling procedure. In addition, as the coherence and sparsity structures have very particular asymptotic behaviour, the performance of compressed sensing varies greatly with dimension, and so does the optimal way of sampling. Fortunately, there is now a developed theory that can guide the user in detail on how to optimise the use of compressed sensing in inverse and imaging problems. I will cover several of the key aspects of the theory accompanied with real-world examples from Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), Surface Scattering, Electron Microscopy, Fluorescence Microscopy etc. Recommended readings: (lectures 1 and 2) - Chapter 4, 6 and 12 in "A mathematical introduction to compressed sensing" (Foucard/Rauhut) - Breaking the coherence barrier: A new theory for compressed sensing - On asymptotic structure in compressed sensing - Structure dependent sampling in compressed sensing: theoretical guarantees for tight frames