The latest developments in computer hardware, sensor technologies, and artificial intelligence can make virtual reality (VR) and virtual spaces an important part of human everyday life. Eye tracking offers not only a hands-free way of interaction but also the possibility of a deeper understanding of human visual attention and cognitive processes in VR. Despite these possibilities, eye-tracking data also reveals users' privacy-sensitive attributes when combined with the information about the presented stimulus. To address all, this survey first covers major works in eye tracking, VR, and privacy areas between 2012 and 2022. While eye tracking in VR part covers the computational eye tracking pipeline from pupil detection and gaze estimation to offline data analysis, for privacy and security, we focus on eye-based authentication as well as computational methods to preserve the privacy of individuals and their eye-tracking data in VR. Later, we outline three main directions by focusing on privacy. In summary, this survey presents an extensive literature review of the utmost possibilities of eye tracking in VR and their privacy implications.