In this work fundamental mechanisms of the interaction between matter and strong shortwavelength pulses have been investigated. For the experiments ultrashort and highly intense light pulses in the extreme-ultraviolet spectral regime from the Free-Electron Laser in Hamburg FLASH have been used. They provide access to new fields of experiments such as imaging of single nanometer-sized structures. Rare gas clusters in the gas phase were investigated as ideal model systems for light-matter interaction. From the scattering patterns of individual clusters, their size and shape as well as the power density they were exposed to can be determined. But also light-induced changes in the clusters on a femtosecond time scale are encoded in the scattered light. By combining single cluster imaging with the coincident measurement of ion spectra, it is possible to sort the single shot data based on the information in the scattering patterns. Thus, the averaging of cluster size distributions and FEL power density profiles which has been apparent in virtually all previous studies on rare gas clusters can be overcome. The well-defined conditions in the single shot data sorted for cluster size yield unique insight into the nanoplasma dynamics.Large xenon clusters with diameters of hundreds of nanometers up to microns could be produced and investigated for the first time. Their scattering patterns reveal a complex, hailstone-like structure, yielding new insight into cluster morphology and growth mechanisms. The characteristic energy distributions of the ions determined from the time-of-flight spectra of very large single xenon clusters indicate that only ions from the outermost atomic monolayer of the cluster explode off. At the same time, the highly efficient recombination in the remaining, quasi-neutral nanoplasma allows for most atoms in the cluster to return back to neutral state. From the corresponding scattering patterns intensity profiles can be extracted. Mie theory yields insight into the optical properties of the clusters. In the intensity profiles of large clusters, modulations are observed which indicate the development of a core-shell system within the nanoplasma. The core and a shell that is up to 50 nm thick differ in the optical properties, yielding insight into ultrafast rearrangements of the electronic structure of the cluster.