Our intention is to develop high-resolution stigmatic spectral imaging in the XUV (2 – 40 nm). We have designed, aligned and tested a broadband stigmatic spectrometer for a range of 12–30 nm, which makes combined use of a normalincidence multilayer mirror (MM) (in particular, a broadband aperiodic MM) and a grazing-incidence plane varied linespace (VLS) reflection grating. The concave MM produces a slightly astigmatic image of the radiation source (for instance, the entrance slit), and the VLS grating produces a set of its dispersed stigmatic spectral images. The multilayer structure determines the spectral width of the operating range, which may amount to more than an octave in wavelength (e.g. 12.5–30 nm for an aperiodic Mo/Si MM), while the VLS grating controls the spectral focal curve. The stigmatism condition is satisfied simultaneously for two wavelengths, 14 and 27 nm. In this case, the condition of non-rigorous stigmatism is fulfilled for the entire wavelength range. A LiF laser plasma spectrum was recorded in one 0.5 J laser shot. A spatial resolution of 26 μm and a spectral resolution of 900 were demonstrated in the 12.5 – 25 nm range. We also report the design of a set of flat-field spectrometers of Harada type with VLS gratings. VLS gratings were made by ebeam and interference lithography. A technique (analytical + numerical) was developed for calculating optical schemes for writing plane and concave VLS gratings with predefined line density variation.