Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional approaches by large margins. In this talk, I will describe the central prevailing themes of this emerging area and a taxonomy that can be used to categorize different problems and reconstruction methods. We will also explore mechanisms for model adaptation; that is, given a network trained to solve an initial inverse problem with a known forward model, we propose novel procedures that adapt the network to a perturbed forward model, even without full knowledge of the perturbation. Finally, I will describe a new class of approaches based on "infinite-depth networks" that can yield up to a 4dB PSNR improvement in reconstruction accuracy above state-of-the-art alternatives and where the computational budget can be selected at test time to optimize context-dependent trade-offs between accuracy and computation.