Analytical tools that have spatial resolution at the nanometre scale are indispensable for the life and physical sciences. It is desirable that these tools also permit elemental and chemical identification on a scale of 10 nm or less, with large penetration depths. A variety of techniques in X-ray imaging are currently being developed that may provide these combined capabilities. Here we report the achievement of sub-15-nm spatial resolution with a soft X-ray microscope--and a clear path to below 10 nm--using an overlay technique for zone plate fabrication. The microscope covers a spectral range from a photon energy of 250 eV (approximately 5 nm wavelength) to 1.8 keV (approximately 0.7 nm), so that primary K and L atomic resonances of elements such as C, N, O, Al, Ti, Fe, Co and Ni can be probed. This X-ray microscopy technique is therefore suitable for a wide range of studies: biological imaging in the water window; studies of wet environmental samples; studies of magnetic nanostructures with both elemental and spin-orbit sensitivity; studies that require viewing through thin windows, coatings or substrates (such as buried electronic devices in a silicon chip); and three-dimensional imaging of cryogenically fixed biological cells.