The service of authentication constitutes the spine of all security properties. It is the phase where entities prove their identities to each other and generally establish and derive cryptographic keys to provide confidentiality, data integrity, non-repudiation, and availability. Due to the heterogeneity and the particular security requirements of IoT (Internet of Things), developing secure, low-cost, and lightweight authentication protocols has become a serious challenge. This has excited the research community to design and develop new authentication protocols that meet IoT requirements. A recent technology, called PUFs (Physical Unclonable Functions), has been the subject of many subsequent publications on lightweight, low-cost, and secure-by-design authentication protocols. This has turned our attention to investigate the most recent PUF-based authentication protocols for IoT. In this paper, we review the security of these protocols. We first provide the necessary background on PUFs, their types, and related attacks. Also, we discuss how PUFs are used for authentication. Then, we analyze the security of PUF-based authentication protocols to identify and report common security issues and design flaws, as well as to provide recommendations for future authentication protocol designers.