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Deep learning, which is a subfield of machine learning, has opened a new era for the development of neural networks. The auto-encoder is a key component of deep structure, which can be used to realize transfer learning and plays an important role in both unsupervised learning and non-linear feature extraction. By highlighting the contributions and challenges of recent research papers, this work aims to review state-of-the-art auto-encoder algorithms. Firstly, we introduce the basic auto-encoder as well as its basic concept and structure. Secondly, we present a comprehensive summarization of different variants of the auto-encoder. Thirdly, we analyze and study auto-encoders from three different perspectives. We also discuss the relationships between auto-encoders, shallow models and other deep learning models. The auto-encoder and its variants have successfully been applied in a wide range of fields, such as pattern recognition, computer vision, data generation, recommender systems, etc. Then, we focus on the available toolkits for auto-encoders. Finally, this paper summarizes the future trends and challenges in designing and training auto-encoders. We hope that this survey will provide a good reference when using and designing AE models. [ABSTRACT FROM AUTHOR]