Image registration is a critical component in the applications of various medical image analyses. In recent years, there has been a tremendous surge in the development of deep learning (DL)-based medical image registration models. This paper provides a comprehensive review of medical image registration. Firstly, a discussion is provided for supervised registration categories, for example, fully supervised, dual supervised, and weakly supervised registration. Next, similarity-based as well as generative adversarial network (GAN)-based registration are presented as part of unsupervised registration. Deep iterative registration is then described with emphasis on deep similarity-based and reinforcement learning-based registration. Moreover, the application areas of medical image registration are reviewed. This review focuses on monomodal and multimodal registration and associated imaging, for instance, X-ray, CT scan, ultrasound, and MRI. The existing challenges are highlighted in this review, where it is shown that a major challenge is the absence of a training dataset with known transformations. Finally, a discussion is provided on the promising future research areas in the field of DL-based medical image registration.