The omnipresence of drones in the civilian air space has led to their malicious usage raising high alert security issues. In this paper, a deep learning approach to detect and identify drones and to determine their flight modes from the remotely sensed radio frequency (RF) signatures is presented. This work intends to detect the presence of drones using two-class classification, the presence along with identification of their make using four-class classification.And this is further extended to the determination of their flight modes using ten-class classification. It employs the proposed architectures of prominent deep learning classifiers, namely, autoencoder (AE), long short-term memory (LSTM), convolutional neural network (CNN), and CNN-LSTM hybrid model. To procure the relevant information from 227 RF signatures having 100 fragments each, the seven significant temporal statistical features, namely, maxima, minima, mean, variance, skewness, kurtosis, and root mean square, are extracted. In a two-class classification scenario, all considered classifiers perform near to idle, whereas in a four-class classification scenario, CNN performs best, followed by AE, CNN-LSTM, and LSTM, respectively. Moreover, in a ten-class classification scenario, AE far outperforms CNN, followed by LSTM and CNN-LSTM, respectively. The best performance in terms of accuracy and classification time confirms the feasibility of the proposed AE classifier for the three considered drone operations.