A comprehensive and well-structured review on the application of deep learning (DL) based algorithms, such as convolutional neural networks (CNN) and long-short term memory (LSTM), in radar signal processing is given. The following DL application areas are covered: i) radar waveform and antenna array design; ii) passive or low probability of interception (LPI) radar waveform recognition; iii) automatic target recognition (ATR) based on high range resolution profiles (HRRPs), Doppler signatures, and synthetic aperture radar (SAR) images; and iv) radar jamming/clutter recognition and suppression. Although DL is unanimously praised as the ultimate solution to many bottleneck problems in most of existing works on similar topics, both the positive and the negative sides of stories about DL are checked in this work. Specifically, two limiting factors of the real-life performance of deep neural networks (DNNs), limited training samples and adversarial examples, are thoroughly examined. By investigating the relationship between the DL-based algorithms proposed in various papers and linking them together to form a full picture, this work serves as a valuable source for researchers who are seeking potential research opportunities in this promising research field.