Toxic language, such as hate speech, can deter users from participating in online communities and enjoying popular platforms. Previous approaches to detecting toxic language and norm violations have been primarily concerned with conversations from online forums and social media, such as Reddit and Twitter. These approaches are less effective when applied to conversations on live-streaming platforms, such as Twitch and YouTube Live, as each comment is only visible for a limited time and lacks a thread structure that establishes its relationship with other comments. In this work, we share the first NLP study dedicated to detecting norm violations in conversations on live-streaming platforms. We define norm violation categories in live-stream chats and annotate 4,583 moderated comments from Twitch. We articulate several facets of live-stream data that differ from other forums, and demonstrate that existing models perform poorly in this setting. By conducting a user study, we identify the informational context humans use in live-stream moderation, and train models leveraging context to identify norm violations. Our results show that appropriate contextual information can boost moderation performance by 35%.