Line-of-sight link blockages represent a key challenge for the reliability and latency of millimeter wave (mmWave) and terahertz (THz) communication networks. To address this challenge, this paper leverages mmWave and LiDAR sensory data to provide awareness about the communication environment and proactively predict dynamic link blockages before they occur. This allows the network to make proactive decisions for hand-off/beam switching, enhancing the network reliability and latency. More specifically, this paper addresses the following key questions: (i) Can we predict a line-of-sight link blockage, before it happens, using in-band mmWave/THz signal and LiDAR sensing data? (ii) Can we also predict when this blockage will occur? (iii) Can we predict the blockage duration? And (iv) can we predict the direction of the moving blockage? For that, we develop machine learning solutions that learn special patterns of the received signal and sensory data, which we call \textit{pre-blockage signatures}, to infer future blockages. To evaluate the proposed approaches, we build a large-scale real-world dataset that comprises co-existing LiDAR and mmWave communication measurements in outdoor vehicular scenarios. Then, we develop an efficient LiDAR data denoising algorithm that applies some pre-processing to the LiDAR data. Based on the real-world dataset, the developed approaches are shown to achieve above 95\% accuracy in predicting blockages occurring within 100 ms and more than 80\% prediction accuracy for blockages occurring within one second. Given this future blockage prediction capability, the paper also shows that the developed solutions can achieve an order of magnitude saving in network latency, which further highlights the potential of the developed blockage prediction solutions for wireless networks.