Cybersecurity is a very emerging field that protects systems, networks, and data from digital attacks. With the increase in the scale of the Internet and the evolution of cyber attacks, developing novel cybersecurity tools has become important, particularly for Internet of things (IoT) networks. This paper provides a systematic review of the application of deep learning (DL) approaches for cybersecurity. This paper provides a short description of DL methods which is used in cybersecurity, including deep belief networks, generative adversarial networks, recurrent neural networks, and others. Next, we illustrate the differences between shallow learning and DL. Moreover, a discussion is provided on the currently prevailing cyber-attacks in IoT and other networks, and the effectiveness of DL methods to manage these attacks. Besides, this paper describes studies that highlight the DL technique, cybersecurity applications, and the source of datasets. Next, a discussion is provided on the feasibility of DL systems for malware detection and classification, intrusion detection, and other frequent cyber-attacks, including identifying file type, spam, and network traffic. Our review indicates that high classification accuracy of 99.72% is obtained by restricted Boltzmann machine (RBM) when applied to a custom dataset, while long short-term memory (LSTM) achieves an accuracy of 99.80% for KDD Cup 99 dataset. Finally, this article discusses the importance of cybersecurity for reliable and practicable IoT-driven healthcare systems.