Deep image mapping networks have been recently applied to solving some inverse problems in imaging due to their good mapping capabilities. However, the greater mapping capability can increase the chance of causing some artificial features when test images differ from training images. Combining image mapping networks with an iterative image recovery that naturally considers imaging system physics is an alternative approach to solving inverse problems. This alternative approach can avoid artificial features, by (back-)projecting the output signals of image mapping networks while considering the imaging system physics. By generalizing the state-of-the-art iterative image recovery algorithm using learned regularizer with iteration-wise image mapping networks, this paper proposes a new recurrent convolutional neural network, Momentum-Net. In addition, this paper investigates the theoretical convergence behavior of Momentum-Net. Numerical experiments show that, for a) denoising low signal-to-noise-ratio images, and b) sparseview X-ray computed tomography, the proposed MomentumNet achieves significantly more accurate and faster image recovery, compared to the state-of-the-art data-driven regularizer or the unsupervised autoencoding regularizer.