Biometric recognition technology has made significant advances over the last decade and is now used across a number of services and applications. However, this widespread deployment has also resulted in privacy concerns and evolving societal expectations about the appropriate use of the technology. For example, the ability to automatically extract age, gender, race, and health cues from biometric data has heightened concerns about privacy leakage. Face recognition technology, in particular, has been in the spotlight, and is now seen by many as posing a considerable risk to personal privacy. In response to these and similar concerns, researchers have intensified efforts towards developing techniques and computational models capable of ensuring privacy to individuals, while still facilitating the utility of face recognition technology in several application scenarios. These efforts have resulted in a multitude of privacy-enhancing techniques that aim at addressing privacy risks originating from biometric systems and providing technological solutions for legislative requirements set forth in privacy laws and regulations, such as GDPR. The goal of this overview paper is to provide a comprehensive introduction into privacy-related research in the area of biometrics and review existing work on Biometric Privacy-Enhancing Techniques (B-PETs) applied to face biometrics. To make this work useful for as wide of an audience as possible, several key topics are covered as well, including evaluation strategies used with B-PETs, existing datasets, relevant standards, and regulations and critical open issues that will have to be addressed in the future.