The real-time discovery of local events (e.g., protests, crimes, disasters) is of great importance to various applications, such as crime monitoring, disaster alarming, and activity recommendation. While this task was nearly impossible years ago due to the lack of timely and reliable data sources, the recent explosive growth in geo-tagged tweet data brings new opportunities to it. That said, how to extract quality local events from geo-tagged tweet streams in real time remains largely unsolved so far. We propose GEOBURST, a method that enables effective and real-time local event detection from geo-tagged tweet streams. With a novel authority measure that captures the geo-topic correlations among tweets, GEOBURST first identifies several pivots in the query window. Such pivots serve as representative tweets for potential local events and naturally attract similar tweets to form candidate events. To select truly interesting local events from the candidate list, GEOBURST further summarizes continuous tweet streams and compares the candidates against historical activities to obtain spatiotemporally bursty ones. Finally, GEOBURST also features an updating module that finds new pivots with little time cost when the query window shifts. As such, GEOBURST is capable of monitoring continuous streams in real time. We used crowdsourcing to evaluate GEOBURST on two real-life data sets that contain millions of geo-tagged tweets. The results demonstrate that GEOBURST significantly outperforms state-of-the-art methods in precision, and is orders of magnitude faster.