Out-of-Domain (OOD) intent detection is vital for practical dialogue systems, and it usually requires considering multi-turn dialogue contexts. However, most previous OOD intent detection approaches are limited to single dialogue turns. In this paper, we introduce a context-aware OOD intent detection (Caro) framework to model multi-turn contexts in OOD intent detection tasks. Specifically, we follow the information bottleneck principle to extract robust representations from multi-turn dialogue contexts. Two different views are constructed for each input sample and the superfluous information not related to intent detection is removed using a multi-view information bottleneck loss. Moreover, we also explore utilizing unlabeled data in Caro. A two-stage training process is introduced to mine OOD samples from these unlabeled data, and these OOD samples are used to train the resulting model with a bootstrapping approach. Comprehensive experiments demonstrate that Caro establishes state-of-the-art performances on multi-turn OOD detection tasks by improving the F1-OOD score of over $29\%$ compared to the previous best method.