In this review, we describe our research on the development of the 13.5 nm coherent microscope using high-order harmonics for the mask inspection of extreme ultraviolet (EUV) lithography. EUV lithography is a game-changing piece of technology for high-volume manufacturing of commercial semiconductors. Many top manufacturers apply EUV technology for fabricating the most critical layers of 7 nm chips. Fabrication and inspection of defect-free masks, however, still remain critical issues in EUV technology. Thus, in our pursuit for a resolution, we have developed the coherent EUV scatterometry microscope (CSM) system with a synchrotron radiation (SR) source to establish the actinic metrology, along with inspection algorithms. The intensity and phase images of patterned EUV masks were reconstructed from diffraction patterns using ptychography algorithms. To expedite the practical application of the CSM, we have also developed a standalone CSM, based on high-order harmonic generation, as an alternative to the SR-CSM. Since the application of a coherent 13.5 nm harmonic enabled the production of a high contrast diffraction pattern, diffraction patterns of sub-100 ns size defects in a 2D periodic pattern mask could be observed. Reconstruction of intensity and phase images from diffraction patterns were also performed for a periodic line-and-space structure, an aperiodic angle edge structure, as well as a cross pattern in an EUV mask.