We present the spectroscopic techniques for studying imaging, spectral, and polarizing properties of multilayer optics intended for solar astronomy and laboratory applications. The measurements were performed using line or quasicontinuous radiation (15 - 400 angstrom) of a point-like source driven by 0.15-J, 5-ns laser pulses at 0.54 micrometer. The imaging quality of focusing normal-incidence multilayer mirrors (MMs) in the subarcsecond resolution range was evaluated from small- source imaging tests employing a high-resolution photographic film. The spectral properties of focusing and plane MMs were measured using the configuration of a transmission grating spectrograph with a medium dispersion, the plate scale typically lying in the range 10 - 30 angstroms/mm. This technique allowed us to measure: (1) the resonance wavelengths versus MM aperture (evaluation of lateral uniformity of the d- spacing); (2) the spectral shape of the primary resonance reflection peak; (3) outside-resonance reflection and higher- order reflection maxima revealed under irradiation by a broad spectrum; (4) for plane MMs, the reflectivities at arbitrary angles of incidence. In the evaluation of plane MMs, the function of focusing radiation was transferred to grazing- incidence toroidal mirrors. About 40 MMs ranging in resonance wavelength from 45 to 310 angstroms, synthesized in different laboratories, were studied using these techniques. A broadband spectrograph comprising a grazing-incidence toroidal mirror and a transmission grating proved to be inherently suited for characterizing the laser-plasma source itself. Two versions of tungsten-coated toroidal mirrors were used, which operated at grazing angles of 7.6 degrees and 4 degrees and had the respective practical short-wave cutoffs at about 40 and 15 angstroms. A source of collimated polarized quasimonochromatic radiation in the 170 to 180 angstrom band was implemented around MMs and our laser-plasma source. The peak polarizance of plane Mo-Si MMs with a d-spacing of 120 angstroms, measured around 175 angstroms at 41 degrees off normal, proved to be 98.2%, which was consistent with the calculated value (98.75%). Two high-resolution high-throughput stigmatic spectrographs were implemented, each comprising a couple of identical concave normal-incidence MMs and a plane grating at grazing incidence. Space-resolved line spectra of laser- produced plasma were obtained and analyzed. The spectrograph for the 130 - 140 angstrom range had a practical resolving power of at least 4000, a plate scale of 0.7 angstroms/mm, and measured 0.6 m. These parameters for the 170 - 190 angstrom range instrument were respectively 24,000, 0.35 angstrom/mm, and 1.1 m. A highly dispersive spectroheliograph was put to a test using a laser-plasma source; the configuration was closely related to that of the spectrograph but involved a reversed ray propagation.