This paper presents a review of deep learning (DL)-based medical image registration methods. We summarized the latest developments and applications of DL-based registration methods in the medical field. These methods were classified into seven categories according to their methods, functions and popularity. A detailed review of each category was presented, highlighting important contributions and identifying specific challenges. A short assessment was presented following the detailed review of each category to summarize its achievements and future potential. We provided a comprehensive comparison among DL-based methods for lung and brain registration using benchmark datasets. Lastly, we analyzed the statistics of all the cited works from various aspects, revealing the popularity and future trend of DL-based medical image registration.