This survey presents a comprehensive review of current literature on Explainable Artificial Intelligence (XAI) methods for cyber security applications. Due to the rapid development of Internet-connected systems and Artificial Intelligence in recent years, Artificial Intelligence including Machine Learning (ML) and Deep Learning (DL) has been widely utilized in the fields of cyber security including intrusion detection, malware detection, and spam filtering. However, although Artificial Intelligence-based approaches for the detection and defense of cyber attacks and threats are more advanced and efficient compared to the conventional signature-based and rule-based cyber security strategies, most ML-based techniques and DL-based techniques are deployed in the black-box manner, meaning that security experts and customers are unable to explain how such procedures reach particular conclusions. The deficiencies of transparency and interpretability of existing Artificial Intelligence techniques would decrease human users' confidence in the models utilized for the defense against cyber attacks, especially in current situations where cyber attacks become increasingly diverse and complicated. Therefore, it is essential to apply XAI in the establishment of cyber security models to create more explainable models while maintaining high accuracy and allowing human users to comprehend, trust, and manage the next generation of cyber defense mechanisms. Although there are papers reviewing Artificial Intelligence applications in cyber security areas and the vast literature on applying XAI in many fields including healthcare, financial services, and criminal justice, the surprising fact is that there are currently no survey research articles that concentrate on XAI applications in cyber security.