We present a survey of methods for assessing and enhancing the quality of online discussions, focusing on the potential of LLMs. While online discourses aim, at least in theory, to foster mutual understanding, they often devolve into harmful exchanges, such as hate speech, threatening social cohesion and democratic values. Recent advancements in LLMs enable artificial facilitation agents to not only moderate content, but also actively improve the quality of interactions. Our survey synthesizes ideas from NLP and Social Sciences to provide (a) a new taxonomy on discussion quality evaluation, (b) an overview of intervention and facilitation strategies, (c) along with a new taxonomy of conversation facilitation datasets, (d) an LLM-oriented roadmap of good practices and future research directions, from technological and societal perspectives.