Intense radiation from lasers has opened up many new areas of research in physics and chemistry, and has revolutionized optical technology. So far, most work in the field of nonlinear processes has been restricted to infrared, visible and ultraviolet light, although progress in the development of X-ray lasers has been made recently. With the advent of a free-electron laser in the soft-X-ray regime below 100 nm wavelength, a new light source is now available for experiments with intense, short-wavelength radiation that could be used to obtain deeper insights into the structure of matter. Other free-electron sources with even shorter wavelengths are planned for the future. Here we present initial results from a study of the interaction of soft X-ray radiation, generated by a free-electron laser, with Xe atoms and clusters. We find that, whereas Xe atoms become only singly ionized by the absorption of single photons, absorption in clusters is strongly enhanced. On average, each atom in large clusters absorbs up to 400 eV, corresponding to 30 photons. We suggest that the clusters are heated up and electrons are emitted after acquiring sufficient energy. The clusters finally disintegrate completely by Coulomb explosion.