Since the 1990s, the use of X-ray phase contrast has been extensively studied for imaging weakly absorbing objects consisting of low-Z elements such as biological soft tissues and polymers. The development of X-ray microscopy was also progressing during this time, although absorption contrast was only available. It was straightforward and important to develop phase-contrast X-ray microscopy. One characteristic in the development is that quantitative phase measurement is possible through the acquisition of phase-contrast images under a specific procedure, thanks to digital X-ray image detectors. Therefore, such a technique is called 'phase imaging' rather than phase-contrast imaging in this review. Highly sensitive three-dimensional phase imaging is feasible in combination with tomography. This article reviews the progress in X-ray phase imaging, especially with regards to X-ray microscopy.