Unlike image-only systems that falter in shadows, glare, and low contrast, LiDAR directly records surface geometry and supports depth-aware quantification. This survey examines LiDAR-based road surface damage classification across the entire pipeline, encompassing acquisition with mobile and terrestrial laser scanning, preprocessing and representation choices, supervised, semi-supervised, and unsupervised learning techniques, as well as multisensor fusion at early, mid, and late stages. A consistent thread is measurement, not just detection: we describe how LiDAR damage classification maps to agency practices such as the Distress Identification Manual and the Pavement Condition Index. We summarize datasets and evaluation protocols for detection, segmentation, 3D reconstruction, and ride quality. We outline practical concerns for corridor-scale deployment: calibration and timing, intensity normalization, tiling/streaming, and runtime budgeting. The review concludes with open problems and outlines directions for robust, severity-aware, and scalable field systems.