This paper provides a review of various machine learning approaches that have appeared in the literature aimed at individualizing or personalizing the amplification settings of hearing aids. After stating the limitations associated with the current one-size-fits-all settings of hearing aid prescriptions, a spectrum of studies in engineering and hearing science are discussed. These studies involve making adjustments to prescriptive values in order to enable preferred and individualized settings for a hearing aid user in an audio environment of interest to that user. This review gathers, in one place, a comprehensive collection of works that have been conducted thus far with respect to achieving the personalization or individualization of the amplification function of hearing aids. Furthermore, it underscores the impact that machine learning can have on enabling an improved and personalized hearing experience for hearing aid users. This paper concludes by stating the challenges and future research directions in this area.