Recent years have seen a surge of interest in dialogue translation, which is a significant application task for machine translation (MT) technology. However, this has so far not been exten- sively explored due to its inherent characteristics including data limitation, discourse properties and personality traits. In this article, we give the first comprehensive review of dialogue MT, in- cluding well-defined problems (e.g., 4 perspectives), collected resources (e.g., 5 language pairs and 4 sub-domains), representative approaches (e.g., architecture, discourse phenomena and personality) and useful applications (e.g., hotel-booking chat system). After systematical investigation, we also build a state-of-the-art dialogue NMT system by leveraging a breadth of established approaches such as novel architectures, popular pre-training and advanced techniques. Encouragingly, we push the state-of-the-art performance up to 62.7 BLEU points on a commonly-used benchmark by using mBART pre-training. We hope that this survey paper could significantly promote the research in dialogue MT.