Older drivers are at a severely higher risk for motor vehicle crash involvement. Due to the global aging of the population, this increased crash risk has a significant impact on society, as well as on an older individual’s quality of life. For this reason, there is a need for understanding how normal age-related changes in cognition and underlying brain dynamics impact driving performance to identify the functional and neurophysiological biomarkers that could be used to design strategies to preserve or improve safe driving behavior in older persons. This review provides an overview of the literature on age-related changes in cognitive functioning and brain dynamics that impact driving simulator performance of healthy persons. A systematic literature search spanning the last ten years was conducted, resulting in 22 eligible studies. Results indicated that various aspects of cognition, most importantly executive function, complex attention, and dual tasking, were associated with driving performance, irrespective of age. However, there was a distinct age-related decline in cognitive and driving performance. Older persons had a more variable, less consistent driving simulator performance, such as more variable speed adaptation or less consistent lane keeping behavior. Only a limited number of studies evaluated the underlying brain dynamics in driving performance. Therefore, future studies should focus on implementing neuroimaging techniques to further unravel the neural correlates of driving performance.