Dialogue summarization aims to condense the original dialogue into a shorter version covering salient information, which is a crucial way to reduce dialogue data overload. Recently, the promising achievements in both dialogue systems and natural language generation techniques drastically lead this task to a new landscape, which results in significant research attentions. However, there still remains a lack of a comprehensive survey for this task. To this end, we take the first step and present a thorough review of this research field carefully and widely. In detail, we systematically organize the current works according to the characteristics of each domain, covering meeting, chat, email thread, customer service and medical dialogue. Additionally, we provide an overview of publicly available research datasets as well as organize two leaderboards under unified metrics. Furthermore, we discuss some future directions, including faithfulness, multi-modal, multi-domain and multi-lingual dialogue summarization, and give our thoughts respectively. We hope that this first survey of dialogue summarization can provide the community with a quick access and a general picture to this task and motivate future researches.