We report the characterization of a scanning transmission X-ray microscope which makes use of an undulator X-ray source, a high resolution scanning stage, and a 45 nm outer zone width, > 10% diffraction efficiency Fresnel zone plate as the probe-forming optic at soft X-ray wavelengths (typically 3.64 nm). The modulation transfer function of the instrument is in good agreement with theory; it remains above 0.1 to spatial frequencies corresponding to structures as small as 25 nm. This spatial frequency limit is the best obtained in any soft X-ray microscope, and is about a factor of ten better than what is attained in conventional or confocal visible light microscopes. By deconvolving the zone plate point spread function from the image, the microscope has been used to resolve 36 nm features in a gold test object. The microscope is used primarily for the study of thick, hydrated biological specimens with minimal radiation dose. © 1991.