Industrial defect detection is vital for upholding product quality across contemporary manufacturing systems. As the expectations for precision, automation, and scalability intensify, conventional inspection approaches are increasingly found wanting in addressing real-world demands. Notable progress in computer vision and deep learning has substantially bolstered defect detection capabilities across both 2D and 3D modalities. A significant development has been the pivot from closed-set to open-set defect detection frameworks, which diminishes the necessity for extensive defect annotations and facilitates the recognition of novel anomalies. Despite such strides, a cohesive and contemporary understanding of industrial defect detection remains elusive. Consequently, this survey delivers an in-depth analysis of both closed-set and open-set defect detection strategies within 2D and 3D modalities, charting their evolution in recent years and underscoring the rising prominence of open-set techniques. We distill critical challenges inherent in practical detection environments and illuminate emerging trends, thereby providing a current and comprehensive vista of this swiftly progressing field.