In the past few years, the commercial use of drones has exploded, since they are a safe and cost-effective solution for many kinds of problems. However, this fact also opens the door for malicious use. This work presents a novel system able to detect and recognise drones from other targets, allowing the police and security agencies to deal with this new aerial thread. The proposed system only uses a persistent range-Doppler radar, avoiding the restrictions of the optical sensors, usually required for the recognition part. The processing is based on constant false alarm rate detection stage, followed by a convolutional neural network that performs the recognition. This network takes as input raw range-Doppler radar data and predicts their class (car, person, or drone). For this purpose, an extensive controlled trial test campaign has been performed, resulting in a novel dataset with more than 17,000 samples of drones, cars, and people, acquired in real outdoor scenarios. As far as authors' knowledge, this is the first range-Doppler radar database for the recognition of drones and other targets. The high-accuracy results (99.48%) suggest that this system could be successfully used in security and defence applications to discriminate between drones and other entities.