LiDAR is one of the most commonly adopted sensors for simultaneous localization and mapping (SLAM) and map-based global localization. SLAM and map-based localization are crucial for the independent operation of autonomous systems, especially when external signals such as GNSS are unavailable or unreliable. While state-of-the-art (SOTA) LiDAR SLAM systems could achieve 0.5% (i.e., 0.5m per 100m) of errors and map-based localization could achieve centimeter-level global localization, it is still unclear how robust they are under various common LiDAR data corruptions. In this work, we extensively evaluated five SOTA LiDAR-based localization systems under 18 common scene-level LiDAR point cloud data (PCD) corruptions. We found that the robustness of LiDAR-based localization varies significantly depending on the category. For SLAM, hand-crafted methods are in general robust against most types of corruption, while being extremely vulnerable (up to +80% errors) to a specific corruption. Learning-based methods are vulnerable to most types of corruptions. For map-based global localization, we found that the SOTA is resistant to all applied corruptions. Finally, we found that simple Bilateral Filter denoising effectively eliminates noise-based corruption but is not helpful in density-based corruption. Re-training is more effective in defending learning-based SLAM against all types of corruption.