Brainwaves, which reflect brain electrical activity and have been studied for a long time in the domain of cognitive neuroscience, have recently been proposed as a promising biometric approach due to their unique advantages of confidentiality, resistance to spoofing/circumvention, sensitivity to emotional and mental state, continuous nature, and cancelability. Recent research efforts have explored many possible ways of using brain biometrics and demonstrated that they are a promising candidate for more robust and secure personal identification and authentication. Although existing research on brain biometrics has obtained some intriguing insights, much work is still necessary to achieve a reliable ready-to-deploy brain biometric system. This article aims to provide a detailed survey of the current literature and outline the scientific work conducted on brain biometric systems. It provides an up-to-date review of state-of-the-art acquisition, collection, processing, and analysis of brainwave signals, publicly available databases, feature extraction and selection, and classifiers. Furthermore, it highlights some of the emerging open research problems for brain biometrics, including multimodality, security, permanence, and stability.