Improving a vehicle driver’s performance decreases the damage caused by, and chances of, road accidents. In recent decades, engineers and researchers have proposed several strategies to model and improve driving monitoring and assistance systems (DMAS). This work presents a comprehensive survey of the literature related to driving processes, the main reasons for road accidents, the methods of their early detection, and state-of-the-art strategies developed to assist drivers for a safe and comfortable driving experience. The studies focused on the three main elements of the driving process, viz. driver, vehicle, and driving environment are analytically reviewed in this work, and a comprehensive framework of DMAS, major research areas, and their interaction is explored. A well-designed DMAS improves the driving experience by continuously monitoring the critical parameters associated with the driver, vehicle, and surroundings by acquiring and processing the data obtained from multiple sensors. A discussion on the challenges associated with the current and future DMAS and their potential solutions is also presented.