Abstract-This paper presents a comprehensive survey of existing authentication and privacy-preserving schemes for 4G and 5G cellular networks. We start by providing an overview of existing surveys that deal with 4G and 5G communications, applications, standardization, and security. Then, we give a classification of threat models in 4G and 5G cellular networks in four categories, including, attacks against privacy, attacks against integrity, attacks against availability, and attacks against authentication. We also provide a classification of countermeasures into three types of categories, including, cryptography methods, humans factors, and intrusion detection methods. The countermeasures and informal and formal security analysis techniques used by the authentication and privacy preserving schemes are summarized in form of tables. Based on the categorization of the authentication and privacy models, we classify these schemes in seven types, including, handover authentication with privacy, mutual authentication with privacy, RFID authentication with privacy, deniable authentication with privacy, authentication with mutual anonymity, authentication and key agreement with privacy, and three-factor authentication with privacy. In addition, we provide a taxonomy and comparison of authentication and privacypreserving schemes for 4G and 5G cellular networks in form of tables. Based on the current survey, several recommendations for further research are discussed at the end of this paper.