Image registration is a fundamental task in multiple medical image analysis applications. With the advent of deep learning, there have been significant advances in algorithmic performance for various computer vision tasks in recent years, including medical image registration. The last couple of years have seen a dramatic increase in the development of deep learning-based medical image registration algorithms. Consequently, a comprehensive review of the current state-of-the-art algorithms in the field is timely, and necessary. This review is aimed at understanding the clinical applications and challenges that drove this innovation, analysing the functionality and limitations of existing approaches, and at providing insights to open challenges and as yet unmet clinical needs that could shape future research directions. To this end, the main contributions of this paper are: (a) discussion of all deep learning-based medical image registration papers published since 2013 with significant methodological and/or functional contributions to the field; (b) analysis of the development and evolution of deep learning-based image registration methods, summarising the current trends and challenges in the domain; and (c) overview of unmet clinical needs and potential directions for future research in deep learning-based medical image registration.