An electroencephalogram (EEG) is a measurement that reflects the overall electrical activity in the brain. EEG signals are effective for biometric authentication and robust against malware attacks and any kind of fraud activities due to the uniqueness of the signals. Significant progress in research on EEG-based authentication has been achieved in the last few years, with machine learning being extensively used for classifying EEG signals. However, to the best of our knowledge, there has been no investigation into the overall progress made in such research. In this paper, the literature on the various factors involved in state-of-the-art biometric authentication systems is reviewed. We provide a thorough comparison of different machine learning biometric authentication techniques. The comparison criteria include the research objectives, machine learning algorithms, computational complexity, source of brainwaves, feature extraction methods, number of channels, and so on. Alongside the discussion of existing works, directions for future research are suggested to improve authentication accuracy. This paper provides an in-depth discussion of different advanced biometric authentication techniques, and a vivid picture of state-of-the-art machine learning-based biometric authentication techniques using EEG.