SAR image targets detection and recognition (SAR-TDR) have become research hotpots in the remote sensing application. These targets include ships, vehicles, aircrafts, oil tanks, bridges, and so on. However, with the rapid development of SAR technology and increasingly complex electromagnetic environment, complex characteristics of SAR images bring severe challenges to the accurate SAR-TDR via traditional physical models or manually features extraction based machine learning methods. In recent years, deep learning (DL), as a powerful automatic features extraction algorithm, has been widely used in the computer vision domain. More specifically, DL has also been introduced into the SAR-TDR tasks and effectively achieved good performances in terms of accuracy, real-time, etc. With the rapid development of DL, SAR images processing, and practical requirements of SAR-TDR in civilian and military domains, it is very necessary to make a systematic survey on the SAR-TDR in past few years. In this survey paper, we mainly conduct a systematic overview of DL-based SAR-TDR literatures on two tasks, i.e., targets recognition (e.g., ground vehicles, ships, and aircrafts) and targets detection (e.g., ships, aircrafts,, change detection, seasurface oil spill, and oil tanks), respectively. More specifically, our related works about these topics are also presented to verify the effectiveness of DL-based methods. Firstly, several DL methods (e.g., CNN, RNN, AEs, and GNN), commonly used to SAR-TDR, are briefly introduced. The systematic review of DL-based SAR-TDR (including our related works) is then presented. Finally, the current challenges and future possible research directions are deeply analyzed and discussed.