A large and increasing number of people around the world experience cognitive disability. Rehabilitation robotics has provided promising training and assistance approaches to mitigate cognitive deficits. In this article, we carried out a systematic review on recent developments in robot-assisted cognitive training. We included 99 articles in this work and described their applications, enabling technologies, experiments, and products. We also conducted a meta analysis on the articles that evaluated robot-assisted cognitive training protocol with primary end users (i.e., people with cognitive disability). We identified major limitations in current robotics rehabilitation for cognitive training, including the small sample size, non-standard measurement of training and uncontrollable factors. There are still multifaceted challenges in this field, including ethical issues, user-centered (or stakeholder-centered) design, the reliability, trust, and cost-effectiveness, personalization of the robot-assisted cognitive training system. Future research shall also take into consideration human-robot collaboration and social cognition to facilitate a natural human-robot interaction.