The vulnerabilities of fingerprint authentication systems have raised security concerns when adapting them to highly secure access-control applications. Therefore, Fingerprint Presentation Attack Detection (FPAD) methods are essential for ensuring reliable fingerprint authentication. Owing to the lack of generation capacity of traditional handcrafted based approaches, deep learning-based FPAD has become mainstream and has achieved remarkable performance in the past decade. Existing reviews have focused more on hand-cratfed rather than deep learning-based methods, which are outdated. To stimulate future research, we will concentrate only on recent deep-learning-based FPAD methods. In this paper, we first briefly introduce the most common Presentation Attack Instruments (PAIs) and publicly available fingerprint Presentation Attack (PA) datasets. We then describe the existing deep-learning FPAD by categorizing them into contact, contactless, and smartphone-based approaches. Finally, we conclude the paper by discussing the open challenges at the current stage and emphasizing the potential future perspective.