The subject of X-ray microscopy (high-resolution X-ray imaging of general nonperiodic structures), an area in which much progress has been made in recent years, is reviewed. The main techniques are briefly described. Achievable performance levels, which for many years were highly speculative, can now be understood with fair accuracy in terms of basic X-ray and specimen properties, and techniques have progressed to the point where actual results are nearing those levels. In terms of specimen size and imaging resolution, X-ray microscopies lie between electron and light microscopy, and are thus suited to imaging extremely large and complex structures; in addition, they demand little or no specimen preparation, and can be used to observe local composition and chemical state as well as structure. Thus X-rays, which have played the leading role in imaging crystallizable materials, may also prove to be highly valuable in the imaging of very large non-crystalline structures. Throughout the treatment, attention is paid to the relationships connecting the subject with X-ray crystallography.