Vision-based early warnings against civil drones are crucial in the field of public security and are also challenging in visual object detection. Because conventional target detection methods built on handcrafted features are limited in terms of high-level semantic feature representations, methods based on deep convolutional neural networks (DCNNs) have facilitated the main trend in target detection over the past several years. Focusing on the development of civil drone-detection technology based on DCNNs, this paper introduces the advancements in DCNN-based object detection algorithms, including two-stage and one-stage algorithms. Subsequently, existing drone-detection methods developed for still images and videos are summarized separately. In particular, motion information extraction approaches to drone detection are investigated. Furthermore, the main bottlenecks in drone detection are discussed. Finally, potentially promising solutions and future development directions in the drone-detection field are presented.