The paper describes the design of a microscope for studying a betatron radiation source based on the PEARL femtosecond laser complex in the SXR and EUV wavelength range. The main optical element of the microscope is a spherical Schwarzschild objective a x5 magnification. The device allows to study the size and spatial structure of the interaction area of laser radiation with matter, at a selected wavelength in the EUV or SXR range with a resolution of delta x=2.75 μm. The operation wavelength (λ=13.5 nm) is set by multilayer X-ray mirrors. Thin-film absorption filters are used to suppress the background component of the signal. Keywords: SXR and EUV radiation, betatron radiation, imaging x-ray optics, SXR microscope.