While sixth-generation (6G) wireless systems are expected to bring about an explosion of accessible user information and novel technologies, along with new threats to terrestrial and non-terrestrial networks, major concerns associated with the development of 6G networks are privacy and security. Reconfigurable intelligent surfaces (RISs), which have recently emerged as promising candidates to support 6G physical platforms, have proved to be capable of boosting security of next-generation wireless systems. However, due to their easy reconfigurability and low cost, RISs are vulnerable to several security threats, and this vulnerability has not yet been thoroughly addressed in previous research. To fill this gap in the literature, in this review, we aim to thoroughly analyze the security challenges affecting RIS-empowered 6G wireless networks. To this end, we review the attributes of RISs that distinguish them from other relevant technologies, such as multiple-input-multiple-output (MIMO), conventional relaying, backscatter communication (BackCom), as well as outline security and privacy attacks in RIS-assisted 6G applications. Our specific focus is on security and privacy threats associated with the use of RISs with different vital 6G technologies, including millimeter wave (mmWave), terahertz (THz), device-to-device (D2D) communication, Internet of Things (IoT) networks, multi-access edge computing (MEC), integrated sensing and communication (ISAC), simultaneous wireless information and power transfer (SWIPT), and non-terrestrial network. The review concludes with an outline of open research challenges and promising future directions to further increase secrecy of RIS-assisted 6G applications. The results of this review contribute to previous research on 6G network security, in general, and RIS-based 6G network security, in particular.