False data injection attack (FDIA) is a common network attack targeting power systems. These attacks can cause abnormal operation of the power system and result in serious economic losses by injecting abnormal data through terminal links and devices which can bypass bad data detection mechanisms. In recent years, deep learning technology has made significant progress in FDIA detection. Deep learning methods can automatically learn and extract attack features with a large amount of data training and powerful learning capabilities. These methods have higher accuracy and robustness than traditional methods. The paper summarizes the research on deep-learning-based power systems FDIA detection technologies in recent years, and covers typical deep learning models such as convolutional, recurrent and graph neural networks, generative adversarial networks, and deep reinforcement learning. First, the FDIA detection principles of various deep learning models and relevant technical methods are analyzed and introduced. Then, a comparative analysis of the above-mentioned technologies in terms of robustness, evaluation indicators, and scalability is proposed to summarize their application scope and shortcomings. Finally, the challenges in current and future research directions are discussed.