While traditional approaches to building damage assessment in the aftermath of natural disasters have relied heavily on time-intensive and costly manual techniques, recent advances in geospatial artificial intelligence (GeoAI) have opened up new possibilities for automating and scaling up this crucial process. Leveraging technologies such as computer vision, remote sensing, and machine learning applied to geospatial data from satellites, drones, and other sensors, GeoAI has the potential to revolutionize how communities assess building damage in disaster-stricken areas and target recovery resources more quickly and effectively. However, efforts to apply GeoAI for building damage assessment also face important challenges regarding data and model quality that require further research. To properly evaluate both the opportunities and challenges of leveraging GeoAI for building damage assessment, this comprehensive review explores the current state of the field through an analysis of recent literature and case studies. An in-depth examination is provided of innovative applications of technologies such as deep learning to high-resolution aerial imagery for automated detection and classification of structural damage. Critical requirements are identified for developing robust GeoAI solutions, such as acquiring comprehensive training data that captures the full range of possible damage patterns and accounting for environmental factors. The review also analyzes efforts by humanitarian organizations and companies to deploy initial GeoAI-powered damage assessment systems in real-world disaster events, highlighting lessons learned.