Rotated object detection (ROD), often termed oriented object detection, is essential for numerous practical tasks, including remote sensing, self-driving systems, urban surveillance, and text recognition in natural scenes. Unlike conventional object detection, ROD must estimate object orientation, making angle regression and loss function design crucial to model performance. This paper presents a comprehensive survey of regression loss functions used in ROD, categorized into coordinate-based, approximated rotated IoU-based, and Gaussian-based approaches. We analyze their theoretical foundations, practical trade-offs, and effectiveness in addressing core challenges including angle periodicity, edge ambiguity, and metric inconsistency. Representative loss functions are benchmarked on standard datasets to evaluate their suitability for various detection frameworks. By emphasizing application contexts such as smart city monitoring and environmental analysis, this survey offers practical guidance for designing robust and efficient ROD systems that support sustainable development goals.