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With the rapid development of the Internet, the massive amount of web textual data has grown exponentially, which has brought considerable challenges to downstream tasks, such as document management, text classification, and information retrieval. Automatic text summarization (ATS) is becoming an extremely important means to solve this problem. The core of ATS is to mine the gist of the original text and automatically generate a concise and readable summary. Recently, to better balance and develop these two aspects, deep learning (DL)-based abstractive summarization models have been developed. At present, for ATS tasks, almost all state-of-the-art (SOTA) models are based on DL architecture. However, a comprehensive literature survey is still lacking in the field of DL-based abstractive text summarization. To fill this gap, this paper provides researchers with a comprehensive survey of DL-based abstractive summarization. We first give an overview of abstractive summarization and DL. Then, we summarize several typical frameworks of abstractive summarization. After that, we also give a comparison of several popular datasets that are commonly used for training, validation, and testing. We further analyze the performance of several typical abstractive summarization systems on common datasets. Finally, we highlight some open challenges in the abstractive summarization task and outline some future research trends. We hope that these explorations will provide researchers with new insights into DL-based abstractive summarization. [ABSTRACT FROM AUTHOR]