Medical image reconstruction aims to acquire high-quality medical images for clinical usage at minimal cost and risk to the patients. Deep learning and its applications in medical imaging, especially in image reconstruction have received considerable attention in the literature in recent years. This study reviews records obtained electronically through the leading scientific databases (Magnetic Resonance Imaging journal, Google Scholar, Scopus, Science Direct, Elsevier, and from other journal publications) searched using three sets of keywords: (1) Deep learning, image reconstruction, medical imaging; (2) Medical imaging, Deep learning, Image reconstruction; (3) Open science, Open imaging data, Open software. The articles reviewed revealed that deep learning-based reconstruction methods improve the quality of reconstructed images qualitatively and quantitatively. However, deep learning techniques are generally computationally expensive, require large amounts of training datasets, lack decent theory to explain why the algorithms work, and have issues of generalization and robustness. The challenge of lack of enough training datasets is currently being addressed by using transfer learning techniques.