The contrast and penetrating power afforded by soft X-rays when they interact with matter makes this form of radiation ideal for studying micrometre-sized objects(1,2). But although soft X-rays are useful for probing detail too fine for visible light microscopy in specimens too thick for electron microscopy, the highest-resolution applications of X-ray imaging have been traditionally limited to crystalline samples. Here we demonstrate imaging (at similar to 75 nm resolution) of a non-crystalline sample, consisting of an array of gold dots, by measuring the soft X-ray diffraction pattern from which an image can be reconstructed. The crystallographic phase problem(3)-the usually unavoidable loss of phase information in the diffraction intensity-is overcome by oversampling(4) the diffraction pattern, and the image is obtained using an iterative algorithm(5). Our X-ray microscopy technique requires no high-resolution X-ray optical elements or detectors. We believe that resolutions of 10-20 nm should be achievable; this would provide an imaging resolution about 100 times lower than that attainable with conventional X-ray crystallography, but our method is applicable to structures roughly 100 times larger. This latter feature may facilitate the imaging of small whole cells or large subcellular structures in cell biology.