Automatic vision-based inspection systems have played a key role in product quality assessment for decades through the segmentation, detection, and classification of defects. Historically, machine learning frameworks, based on hand-crafted feature extraction, selection, and validation, counted on a combined approach of parameterized image processing algorithms and explicated human knowledge. The outstanding performance of deep learning (DL) for vision systems, in automatically discovering a feature representation suitable for the corresponding task, has exponentially increased the number of scientific articles and commercial products aiming at industrial quality assessment. In such a context, this article reviews more than 220 relevant articles from the related literature published until February 2023, covering the recent consolidation and advances in the field of fully-automatic DL-based surface defects inspection systems, deployed in various industrial applications. The analyzed papers have been classified according to a bi-dimensional taxonomy, that considers both the specific defect recognition task and the employed learning paradigm. The dependency on large and high-quality labeled datasets and the different neural architectures employed to achieve an overall perception of both well-visible and subtle defects, through the supervision of fine or/and coarse data annotations have been assessed. The results of our analysis highlight a growing research interest in defect representation power enrichment, especially by transferring pre-trained layers to an optimized network and by explaining the network decisions to suggest trustworthy retention or rejection of the products being evaluated.