Millimeter wave (mmWave) and sub-terahertz communication systems rely mainly on line-of-sight (LOS) links between the transmitters and receivers. The sensitivity of these high-frequency LOS links to blockages, however, challenges the reliability and latency requirements of these communication networks. In this paper, we propose to utilize radar sensors to provide sensing information about the surrounding environment and moving objects, and leverage this information to proactively predict future link blockages before they happen. This is motivated by the low cost of the radar sensors, their ability to efficiently capture important features such as the range, angle, velocity of the moving scatterers (candidate blockages), and their capability to capture radar frames at relatively high speed. We formulate the radar-aided proactive blockage prediction problem and develop two solutions for this problem based on classical radar object tracking and deep neural networks. The two solutions are designed to leverage domain knowledge and the understanding of the blockage prediction problem. To accurately evaluate the proposed solutions, we build a large-scale real-world dataset, based on the DeepSense framework, gathering co-existing radar and mmWave communication measurements of more than $10$ thousand data points and various blockage objects (vehicles, bikes, humans, etc.). The evaluation results, based on this dataset, show that the proposed approaches can predict future blockages $1$ second before they happen with more than $90\%$ $F_1$ score (and more than $90\%$ accuracy). These results, among others, highlight a promising solution for blockage prediction and reliability enhancement in future wireless mmWave and terahertz communication systems.