Recent advances in deep learning, and especially the invention of encoder-decoder architectures, has significantly improved the performance of abstractive summarization systems. While the majority of research has focused on written documents, we have observed an increasing interest in the summarization of dialogues and multi-party conversation over the past few years. A system that could reliably transform the audio or transcript of a human conversation into an abridged version that homes in on the most important points of the discussion would be valuable in a wide variety of real-world contexts, from business meetings to medical consultations to customer service calls. This paper focuses on abstractive summarization for multi-party meetings, providing a survey of the challenges, datasets and systems relevant to this task and a discussion of promising directions for future study.