This paper is aimed at presenting a systematic review of the recent studies on the use of deep learning techniques in aircraft engine health management for fault detection and performance enhancement. This comes in the light of increased growth in the air transport industry, the efficiency of aircraft engines and the efficiency of their maintenance have emerged as key issues in the safety and efficiency of air transport. This paper also identifies some of the key issues emerging from this area such as the data issues and the need to develop very accurate models for faulting and performance enhancement. Some of the most advanced techniques, namely the Long Short-Term Memory (LSTM) networks and Convolutional Neural Networks (CNNs) are discussed as potential solutions to these challenges. The theoretical descriptions of the working principles, as well as the strengths and weaknesses of the methods in question with regard to the problems of aircraft engine health management, are elaborated. Furthermore, this paper introduces the application of Reinforcement Learning as a new technique in the area of optimal control of engine parameters with the hope of improving the dependability and maintainability of aircraft engines and in the process of achieving sustainable growth in the aviation industry. The issues on model interpretability, model generalization and model deployment are also discussed and research guidelines for the future work in this field are also outlined. In this paper, the state of the art and the importance of deep learning as well as the applications of deep learning in aircraft engine health management are discussed based on the literature and case studies.