The necessity for high-speed and low-latency connectivity of the vast number of mobile users is rising with the immense usage of mobile applications. A cloud radio access network (C-RAN) is a promising framework for next-generation cellular communication, which can satisfy the requirements of significantly increasing data traffic and user demands. In C-RAN, the data processing unit can be centralized and virtualized in data centers and can be shared among distributed base stations. Deep learning (DL) appears to be a feasible approach for facilitating the data processing capability, resource management in the cloud, and predicting dynamic traffic in cellular communication. The convergence of C-RAN and DL is expected to bring new prospects to both interdisciplinary research and industrial applications. In this regard, different approaches have been proposed for DL-based C-RAN in the literature. This article provides a comprehensive survey of the state-of-the-art DL techniques applied in C-RAN. A brief introduction of the C-RAN architecture and DL techniques is given to provide insights into these two emerging technologies. Existing surveys are also discussed to highlight the research gap. The reviewed works are categorized into power consumption optimization, network performance maximization, and QoS maximization based on their optimization objectives. The key ideas of DL applied in the reviewed schemes are also mentioned, and the performance evaluation techniques used in the research are discussed and compared. Lastly, research challenges and open research issues are highlighted to provide future research directions.INDEX TERMS Cloud radio access network, deep learning, evaluation techniques, optimization objectives, performance metrics.