Structural disaster damage detection and characterization is one of the oldest remote sensing challenges, and the utility of virtually every type of active and passive sensor deployed on various air- and spaceborne platforms has been assessed. The proliferation and growing sophistication of unmanned aerial vehicles (UAVs) in recent years has opened up many new opportunities for damage mapping, due to the high spatial resolution, the resulting stereo images and derivatives, and the flexibility of the platform. This study provides a comprehensive review of how UAV-based damage mapping has evolved from providing simple descriptive overviews of a disaster science, to more sophisticated texture and segmentation-based approaches, and finally to studies using advanced deep learning approaches, as well as multi-temporal and multi-perspective imagery to provide comprehensive damage descriptions. The paper further reviews studies on the utility of the developed mapping strategies and image processing pipelines for first responders, focusing especially on outcomes of two recent European research projects, RECONASS (Reconstruction and Recovery Planning: Rapid and Continuously Updated Construction Damage, and Related Needs Assessment) and INACHUS (Technological and Methodological Solutions for Integrated Wide Area Situation Awareness and Survivor Localization to Support Search and Rescue Teams). Finally, recent and emerging developments are reviewed, such as recent improvements in machine learning, increasing mapping autonomy, damage mapping in interior, GPS-denied environments, the utility of UAVs for infrastructure mapping and maintenance, as well as the emergence of UAVs with robotic abilities.