## 1 Introduction and Previous Achievements  
Advancements in nanoscience and nanotechnology require tools and methods for nanometer scale resolution imaging. Affecting the matter on nanoscale and subsequent observation of those manipulations is a key factor in the rapid development of nanotechnology. Among the methods employing visible light radiation [1; 2] to image matter at nanometer scales shorter wavelengths are also used, for example in extreme ultraviolet (EUV) and soft X-ray (SXR) microscopy experiments [3]. EUV and SXR microscopy has been proven as useful tools for imaging of objects with nanometer spatial resolution, having advantages such as obtaining additional information about the objects investigated, providing at the same time high optical contrast in the specific wavelength ranges used for this purpose. The main characteristics of the EUV radiation are its strong absorption in very thin layers of materials, so it provides a good optical contrast in thin films and layers of materials. On the other hand, SXR radiation--specifically in the so-called \"water-window\" (\(\lambda\) = 2.3-4.4 nm)--proved to be particularly useful for high resolution imaging of biological samples, because of high optical contrast existing between carbon and water present in those samples. Imaging in short wavelength ranges requires a high photon flux. It is the main reason why, until the development of compact sources, most of the studies in the field of EUV and SXR microscopy were conducted using synchrotron and free-electron laser facilities [4; 5; 6]. Such facilities are still being used not only for the purpose of high resolution imaging, but also for cutting-edge scientific experiments, providing the highest available photon flux, with the possibility to tune the generated wavelength and producing beams with adjustable spatial and temporal coherence. Nevertheless, these sources present some drawbacks because of their high complexity, high maintenance costs, requirement for highly trained staff present during the experiments and, finally, limited user access. Thus, to mitigate those difficulties, the recent progress in development of a new generation of compact EUV and SXR compact sources was made over the last few decades. Nowadays, there are sources which can perform certain experiments, previously restricted to large scale facilities, on a much larger scale in smaller laboratories worldwide. These experiments include imaging with nanometer spatial resolution. Laser-produced plasma sources, emitting short wavelength radiation in the SXR and EUV spectral ranges, offer an important alternative to be the drivers dedicated for compact imaging systems [7; 8]. They can overpass the limited accessibility of large facilities, maintaining a comparable spatial resolution [9; 10]. Over the last few years, many efforts have been made to perform nanometer spatial resolution imaging in the EUV and SXR spectral ranges employing both large scale and compact sources. The synchrotron radiation at \(\lambda\) = 2.4 nm was used for imaging frozen-hydrated samples at atmospheric pressure [11] to observe internal details of algae with a spatial resolution of \(\sim\)35 nm. Using the Advanced Light source synchrotron facility, a 10 nm spatial resolution imaging was obtained at 1.75 nm wavelength both in full-field and in scanning mode [12]. Recently, a novel approach was demonstrated, so-called dark-field X-ray microscopy, which is a non-destructive microscopy technique for the three-dimensional mapping of orientations and stresses [13]. Such a technique allowed multiscale imaging with a spatial resolution of \(\sim\)100 nm. The various capabilities of full-field transmission X-ray microscopy, 3D X-ray tomography, Zernike phase contrast, quantification of absorption, and chemical identification via X-ray fluorescence and X-ray absorption near edge structure imaging is now also possible with synchrotron light [14] for characterization of biomaterials.