With the ever-increasing demand for sub-10 nm integrated circuit chips in the fields such as consumer electronics, interconnect hardware, and electronic medical equipment, the impact of wafer defects introduced by semiconductor manufacturing equipment on the yield and price of integrated circuits will continue to emerge, which makes the manufacturing process control such as the high-speed identification, localization, and classification of typical defects more challenging. Although conventional wafer defect inspection methods such as bright-field, dark-field, and electron-beam imaging can cover most defect inspection scenarios, they cannot balance inspection accuracy, sensitivity, and speed. Emerging techniques such as nanophotonics, computational imaging, quantitative phase imaging, optical vortex, multi-beam scanning electron microscopy, thermal field imaging, and deep learning have shown great potential in improving defect sensitivity, resolution, and contrast, which opens up new possibilities for wafer defect inspection. Hence, we make a comprehensive review for the progress in wafer defect inspection from three aspects: the assessment of defect detectability, the diverse inspection methods and prototypes, and the advanced post-processing algorithms. It is expected to help both researchers and interdisciplinary workers who are new entrants in the field.