Learning-based methods, and in particular deep neural networks, have emerged as highly successful and universal tools for inverse problems such as image and signal recovery and restoration. They achieve state-of-the-art results on tasks ranging from denoising, compression, and image reconstruction from few and noisy measurements. They are starting to be used in important imaging technologies, for example in GEs newest computational tomography scanners and in the newest generation of the iPhone. In the first part of the talk, we give an introduction to image reconstruction with neural networks, and contrast those emerging methods to classical sparsity based recovery approaches, which have been widely studied in the Information Theory community. In the second part of the talk, we focus on image reconstruction with untrained neural network. Untrained networks require no training, and come with recovery guarantees paralleling those of classical approaches. The field has a range of theoretical and practical questions that remain unanswered. In particular, learning and neural network-based approaches often lack the guarantees of traditional physics-based methods, and can make drastic reconstruction errors.