Tweets are the most up-to-date and inclusive stream of in- formation and commentary on current events, but they are also fragmented and noisy, motivating the need for systems that can extract, aggregate and categorize important events. Previous work on extracting structured representations of events has focused largely on newswire text; Twitter's unique characteristics present new challenges and opportunities for open-domain event extraction. This paper describes TwiCal-- the first open-domain event-extraction and categorization system for Twitter. We demonstrate that accurately extracting an open-domain calendar of significant events from Twitter is indeed feasible. In addition, we present a novel approach for discovering important event categories and classifying extracted events based on latent variable models. By leveraging large volumes of unlabeled data, our approach achieves a 14% increase in maximum F1 over a supervised baseline. A continuously updating demonstration of our system can be viewed at http://statuscalendar.com; Our NLP tools are available at http://github.com/aritter/ twitter_nlp.